To directly answer your question, no, do NOT by any means cancel the cards that you don't want now and apply for others after doing so. That could really mess you up big time, and cause your credit score to drop so low that you may not be approved for any credit card afterwards. While on some credit reports it may take 2 to 3 months for a new credit card to show up...I've had one show up within days of me applying for it - that's how I knew my credit limit before I even got the new credit card in the mail. It showed up on my credit report that quick. So while that's not usually normal...I'd imagine it could disappear and/or show as cancelled on your credit report, for each of those cards. Thus...lowering your credit score due to lack of 'active' accounts, and making your overall utilization look higher (since more cards made it look lower).
The cards you have now need to stay active and keep your credit score up, which is what most credit card companies you apply to, look for. That, and of course, your income (if you live with anyone else in the household, I recommend putting down combined household income...they usually note that you can do that on the application, and you'll get a higher credit limit on the cards you apply for; which can make your overall utilization low and your credit score a little higher in doing so).
So...keep the old cards, until you have applied and have been approved for the new cards. Then you can cancel the old ones, if you want. It is good to have one of each type of card...Visa, AmEx, Mastercard, and Discover. Though I personally dislike Mastercard. A lot of places don't seem to accept Discover cards, due to the charges that they get charged by the credit card company, for accepting the card as a payment. It's like 3% of the transaction per person, for them...so not every place takes Discover. The same with AmEx...but they are more widely accepted.
So, bottom line...get those new cards first, then cancel the old cards. Also...my other advise to you is find out ahead of time exactly which cards you want to apply for. Apply for them back-to-back...one right after the other, at the same time. Apply online and most major companies approve you within 60 seconds, and you instantly get the credit limit approved for and know that the card is coming in the mail in 7-10 days. Don't do one on this day and one on that day...do them at the same time, so that you don't have to worry about any drop in credit score between applications - which can happen when you introduce new credit into the scene. That way, you have a better chance of getting approved for all of them, with the highest credit limits. I've done this, it works well. I also say this because each time a credit card company checks your credit for a new card - they can pull a "hard" credit report, which is visible to other companies and affects your credit score. So if you apply for them all at the same time...theoretically, you have a better chance of getting every one approved before any drop in score or any hard inquiries show up on the credit report.
I highly recommend not listening to that Darren person who posted above. While some of what he says seems to make sense...those "live by cash alone" people have no credit and/or poor credit. Sure, they have no credit cards and maybe no debt. But they don't have the credit to get an apartment, get some jobs (some require good credit to hire you), get a car loan, or even a mortgage. These things go by your credit score and utilization.
3 years ago, I had no credit. I started out with a Secure Visa Gold through my bank. It started with $500 credit limit, and after a year, went to $1,000. Now, 3 years later...I have a Platinum Plus Visa through that bank (I upgraded it) with a $15,000 credit limit. I have an AmEx Clear card with a $20,000 credit limit. A Discover More card with a $6,000 credit limit, and a newly applied for Bank of America Accelerated Rewards American Express with a $5,000 credit limit. 4 cards...that's all I need. In total, on two cards...I have a total of about $2,500 charged. The others...I just let them sit there and use them every once in a while for a small purchase, to keep my credit and the card itself active. I try to stick with one or two main cards for all of my spending, that way I don't have a bunch of payments with a lot of interest going here and there to each card; just a couple. I just added the last credit card on as a last resort to start building on, in case one of the companies does something crazy (like dropping credit limits randomly, as some companies are doing to some people due to the 'credit crunch' now). My credit score should be around 780 to 800 now, and the last time I checked it, the credit score description stated that I had better credit than about 50% of the U.S. So...I said those things above to give you an idea of how quickly things impact credit reports and credit scores, for better or for worse.
I would highly recommend consolidating down to 4 cards. Having that many credit cards (7) can look bad, and I realize that you DON'T want that many. People think my limits are kind of high...but it's only 4 cards that I have. It looks better than the 12 to 50 that many people have. Yes, I've worked in Retail and had a lady pull out a BOOK of like 50 cards and she had to browse through them to find the right card. She was probably in debt up to her ears.
What type of cards do you currently have? Are any of the current 7 cards that you have, major credit cards? If so...you can upgrade to a better card/APR rate on a higher tier. While what the other person told you is true about it not being too bad by keeping open your current cards...I see what you want to do, and why you want to consolidate. If only one or two (or none) of those 7 cards that you already have are major credit cards (Visa, Mastercard, Discover, AmEx)...that doesn't look as good on your credit report. Major companies are highly favored with most creditors, and therefore 4 major cards are better than 7 or 11 (if you kept the old ones as suggested by the other person). Avoid any credit cards with annual fees - that's the bottom line before applying to anything. They already charge interest - why should anyone pay them a fee just to have the card?
Realistically, if you are not planning on buying a car or getting an apartment/mortgage anytime soon, then it doesn't matter and you can drop the 7 cards for 4 major cards (unless some of those are already major cards...in that case, upgraded them to a higher tier. E.g. Gold to Platinum, etc.). Yes, your credit score will drop some by cancelling the other older cards...even majorly. But it will go right back up with a little time, and with good payments on the 4 major cards. Focus on those 4 cards...make the good payments each month (always pay more than the minimum amount...it builds a better history with the company). Use them for typical purchases you would make with cash and pay them off at the end of the month (therefore, not having to pay interest since you aren't carrying a balance over that monthly period). It's like an interest-free loan for a month. If you would use cash anyway, then just put what you'd buy with cash on your credit cards, and pay it off each month. Ask for credit limit increases when possible (the length of time between each increase varies...so you have to find that out from the company themselves). Higher credit limits on 4 cards is better than lower credit limits on 7 or 11 cards.
There IS such a thing as too much credit...but that depends on your income and available credit is NOT debt by any means. Just make sure that you don't get in over your head. I'd drop down to the 4 major credit cards. You'll be happy with knowing that they are 'major' credit cards, and you'll have just 4 to make good payments to and get increases on. The others...that's just too many to carry around and keep up with in general. With only a $20,000 annual income...I have total credit that is $46,000 spread over 4 major credit cards. That's not 'bad' or 'too much' credit. It looks good...but of course, I have to keep that under control and make sure that I don't overcharge on the credit cards. I think of credit cards as...whatever I put on them, is going directly out of my bank account. Too many people get into the "charge it" mentality, as if they don't have to pay it back or they'll "pay it back later when they have more money"...and that's what gets them in trouble.