I suspect you do not want to hear this, but no, you should not apply for more credit cards. You need to close anything with a monthly fee (including the merrick you just opened). You realize that merrick wants $264/year in monthly fees for a $500 credit limit? Keep the capital one, hsbc and best buy. If the fingerhut has no fees, keep that as well. There is no benefit, from a rebuilding standpoint, in having more than 2-3 cards open. The credit limits really don't matter. Pay off (all the way off) all but one card. Allow the final card to post a balance no more than 9% of your available credit (and no more than 20% of the limit of that card).
If you are serious about wanting a mortgage in 6 months, you should not apply for anything else between now and then. Instead, devote all your energy to trying to repair your credit report. Getting from the 560s to the 640s in 6 months is going to be a challenge and adding inquiries/new accounts will make it harder.
If you are thinking more 1-2 years from now, I'd cancel all but the two cards with the lowest annual fees, take all the money you would have spent on monthly/annual fees/interest and open a USAA secured card (I'd be shocked if they approved you for an unsecured card with your scores, so wouldn't waste the inquiry). Use it sparingly and pay it off each month. Do that for a year, repair what you can from your credit report, don't carry balances and don't apply for anything else. In a year, you should be able to get above 640, qualify for a better mortgage and, once you close, start applying for prime cards.
Closing cards will not impact your AAoA, applying for new cards will. Cap 1 and HSBC won't close your accounts for minimal use, they know their cards are for rebuilding. As long as you pay the annual fee, they will be happy.
In wallet: Ink Plus 10k, AMEX TE 25k. In bag: CSP 16k, USAA WMC 15k, Hyatt 13k, United MPE 12k, AMEX HHonors 3k. In SD: Cap 1 QS 5k, Discover IT 7k. FICO 08 says my EQ is now 844, was 510 in 2010.
