No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
I'm in pretty much the same boat as you. Lots of lates on student loans, no collections, etc. You've been at it about 6 months longer than I have, though.
I sadly don't have any suggestions for you; I've been harassing Nelnet with goodwill requests by email, and just yesterday got a call from someone who tried to tell me that I should stop emailing their executives every day because they're not going to GW the lates (Nelnet is full of **bleep**). I'm going to ignore this, and hope I annoy them into doing it over time. Otherwise I'll have to wait the full 7 years for them to come off, considering my last late was in March of '16.
Good luck to you, though, and I'll make sure to pass on any info I come across if I do.