HI Jhouser - welcome to the forums!
Nice move finding a way to significantly pay down your debt without taking on new debt - you should feel really good about that. I think you should see some significant gains in your scores once your new balances update. We'd be better equipped to guesstimate score improvement if we knew the credit limits and balances of all your revolving accounts though. Points are gained from paying down debt when the paydown causes your overall utilization to drop below a scoring threshold; the thresholds are as follows (high to low, red is too high): 88.9%, 68.9%, 48.9%, 28.9%, 8.9%.
We typically see ~10 point gains for each threshold crossed upon paydown (conversely, we see ~10 point losses for each threshold crossed when debt is increased). You can calculate where your util started and where it will be once updated by taking the total of your credit balances and dividing that by the total of your credit limits, before and after paydown.
Getting that charge-off account settled is very good for your scores -- (1) the outstanding balance is being factored into your utilization and because the account was charged-off, the credit limit was $0, so the utilization was reporting over 100% -- which is a big scoring negative. Now that you've settled, the account balance will be updated to $0, reducing utilization; (2) now that the account has been paid, it will officially close preventing any future updating of negative information by the original creditor which will allow the derogatory to age and hurt a bit less over time - although it will still hold your scores down until it falls off your reports, your scores will continue to rise despite the presence of the charge-off.
I'd wait for your accounts to fully update on your reports before seeking a prequal for an auto loan - but I think you'll be OK with a paid charge-off and a few late pays... though your interest rate may be a tad high. And-- if this will be your only active (open) installment loan, you'll likely see a nice boost to your scores once it reports because you'll be fulfilling a piece of the FICO 'credit mix' requirement.
Definately begin a Goodwill campaign to have those late pays removed. For now, just work paying off your two active cards, get your auto loan and let that age a bit, and just keep paying everything on-time -- try to get into the habit of paying your statement balances in full each month whenever possible to prevent the accumulation of debt. Try to keep overall utilization below 8.9% and individual card utilization below 28.9% of their respective credit limits for optimal scoring and you'll be fine.
My score was a 566 before making these moves.
Also want to make sure you are referencing FICO scores and not Vantage 3 (ie. Credit Karma, WalletHub, Credit Sesame, etc.) scores.
FICO scores are the scores you want to monitor for improvement as they are the ones used by the overwhelming majority of lenders to make credit decisions, not Vantage 3. FICO and Vantage scores also respond differently to changes in your credit so it is common to see a fairly large gap between to two types of scores for the same credit profile.