Are you saying that you have like late payments on these cards? When you close an account, it can remain on your report for up to 10 years. I've seen some people mention that their closed cards stayed on longer, and some were removed within a few years, so it's very YMMV. However, closing the account doesn't erase things like late payments. Those will remain on your report for as long as the account is on there.
As for the annual fees, you just need to cancel the cards before the annual fee hits again. If you were already charged for the year, then there's no sense in closing it, unless you REALLY don't need/want it. However, having cards open helps with your utilization, which is a big factor in your credit score.
EDIT:
Ok I think I read that first sentence wrong. The only reason your score would drop from closing a card is because it lowered your utilization. Closing the card removes your available credit and if you have balances on other cards, it can definitely hurt. Fixing that would simply require you to pay off any balances and keep your overall utilization below 8.9%. So, essentially, closing your cards will only lower your scores if it makes your utilization go up.
@Anonymous wrote:No, no lates on any of these. They are all in good standing... it's just the annual fees I don't want to keep paying now that I have cards without them. These were my rebuilding cards.
Yeah, I read the first sentence wrong. I edited my response up above. 
@Anonymous wrote:When you close an account - generally, how long does it drop your score before it gets better?
Also if I close more than one, how is it affected? I've recently been able to get someone of the better cards I was wanting and now I want to get rid of the ones I have with these pesky annual fees before they hit again! I tried to ask for them to waive them but they would not.
These include: Surge Mastercard, CreditOne Bank, First Premier & Lend Up Arrow
Closing the accounts doesn't drop your score at all.
@Anonymous wrote:Closing accounts does not drop FICO scores. This is one of the top misconceptions on this forum. I think we need to make a list of the top 10 misconceptions and have it pinned as a sticky thread.
Yeah but what happens when we get to 11, 12, and 13?
@Anonymous wrote:Either:
A - They are included in the list and the list can be as long as it needs to be.
or
B - We deem that anything past a certain point (say, 10) doesn't come up often enough to be considered a top misconception.
I was just trying to be humorous. My way of saying that there seem to be an infinite number of misconceptions floating around