It's unfortunate that you settled the collection rather than doing a pay for delete. A PFD would have gotten the collection removed entirely. The old mortgage models do not reward you for paid collections in the same way that (say) FICO 9 does. A paid collection on your reports is still punished quite a bit.
The answer to your question about paying down your installment debt is not entirely simple. That is because it is possible that FICO might score student loan differently (separately) from other kinds of installment debt. Possible, not certain. Right now we do not have an absolute certain answer to that, though we hope to get an answer that this year with some carefully crafted test scenarios.
Let's suppose that FICO does not score SL debt differently from (say) auto or other kinds of installment debt. This is probably the case -- but as I say we do not know yet. In this case, it means that your HUGE SL debt will dominate all your other loans. Paying the car loans down won't help you, because your total aggregate Amount Owed compared to the Original Amount Borrowed of all your loans will still be so high.
But... let's suppose in contrast that FICO does score the two kinds of debt separately. In that case, paying your other non-SL loans down quite a bit, while still keeping them open, might well give you the scoring bonus you are looking for.
The problem with trying this second thing (hoping that the FICO mortgage models score SL differently and so choosing a paydown strategy) is that (a) we do not know that the mortgage models score SL differently and (b) if they don't score it separately, then you will have wasted a bunch of cash that you might have used toward a down payment.
This is why I was trying to find out if there were other ways I could help you, rather than the paydown method.
