I have a good credit score, but I've been really lucky (and fairly responsible due to anxiety and paranoia) to have no missed payments or derogatory marks. In that aspect I can't give much advice although it seems like you are taking the right steps in doing what you can to get marks removed as quickly as possible.
What I can tell you is I have found that utilizing more than 75% to 90% of my available credit card limits greatly affects my credit score.
I pay off my store cards in full every month, but my main card had steadily accrued a balance of over $3,000 from the time I opened the card in college about 6 years ago. I made some big purchases and ended up with a total of $5000 on that card. That one card having that high of a balance utilization made my credit score drop to between 650 and 695 each time I checked it (as the balance wavered between 4 and 5k). I transferred that entire card balance into a direct personal loan with a fixed interest rate and my credit score shot up to 750 once the reporting agencies registered it. I am paying it off aggressively now so I'm hoping my score will tick up a little more as I continue to pay off the personal loan. If I'm reading your post right did you just pay off that card? If you did it in one big chunk, give it a month or so and see if that didn't help your score some.
Credit Karma was always yelling at me for using 90% of my available credit. Credit utilization has a high impact along with payment history & derogatory marks (so the one you probably have the most control over at this point is your utilization). Hard inquiries do not seem to have a negative impact on my scores but I have found that having too many inquiries in a short period of time has prevented me from being able to open new lines of credit, which sucks because I've moved a lot and every rental agency runs my credit.
I will say every time my bank or a rental agency runs my credit they've shown I have a much higher score than what Credit Karma shows me. I think Experian is a little more forgiving which is the one major agency CK doesn't report for.
(FYI my credit is all based on CC and one federal student loan I do not have a car or home loan to boost it)
I wish you luck. If your student loans are in forbearance you might try to get on a income-based repayment plan so you can start making payments when you've got your cards paid off. Making timely payments on debts will help build your score up, although I can't say how much.