Mortgage are underwritten with an old FICO model. We call them mortgage scores because most mortgages use them. But they don't emphasize mortgages more than other FICO models. I assume the FICO 08 Mortgage enhanced scores would emphasize past mortgage history, but I have never seen that model used.
You are doing the right thing in trying to get rid of your baddies. Until the last one is removed, your score will suffer. There are a couple of other things you can do, though.
1) If you have a SO with good credit card history, have them add you as an AU on their good cards.
2) Only let one credit card report a balance. You want it to report a balance between 1-9% of the credit limit on that card.
3) Try to get your overall installment utilization as low as possible. It will only help your EX-98 score, but every little bit helps. If you do not have any open installment loans, consider opening a small shared secured installment loan for $500. Then immediately pay off all but $50. You can find threads here about the pros and cons of doing so.
The above can help a little with your scores, but you will probably hit a ceiling with any major baddies on your report.
