I'm not familiar with the BofA service, but I really doubt they're giving you FICO scores. If the service is costing you anything, I would frankly get rid of it.
Now, if you want to improve your credit, you need to actually get some more credit and build positive payment history again. A
secured credit card can serve that purpose if your FICO scores are in the bad credit range and you can't get access to a regular credit card. If you have a spouse/close relative that's willing to add you as an AU on one of their oldest credit cards with a low credit utilization ratio, that's another option that should help boost your credit scores.
Also, pay down that $508 balance on your existing card and your scores should improve quite a bit. Your CU ratio accounts for about 30% of your FICO scores and you want to ideally keep your ratio under 10%. This is the very first thing you should do, before you even think about adding a new credit card to the mix.
You can't remove credit inquiries unless they're bogus and the creditor didn't have a valid reason for pulling your credit. Don't worry about any of the "soft" inquiries- they're not hurting your FICO scores anyway.
Lastly, dispute any late payments that you believe to be incorrect through the CRAs. If the late payments were legitimate and the accounts are closed, I doubt you'll be able to get them removed.
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