Welcome to the forum. I'm not sure I can give you specifics on what's driving your auto score lower. I have some ideas but with additional information myself or others on the forum will be able to help.
The 65 percent utilization is definitely hurting your scores along with the late payment history. When you reduce your card to 29;percent utilization be sure to reduce it enough to allow for the interest expense and still be below 29;percent.
A few questions
1) What is the source for your initial scores of 635 and 650? Are these Fico scores? Equifax, Experian or Trans Union
2) For the auto scores, were these Fico scores? What version? Fico 8, 9?
3) For the lates, how long since the last late?
4) What's the worst late? 30, 60, 90?
5) Any new inquiries in the past year? How new is the newest?
6) Any new accounts in past year? How many months?
7) How many credit cards? What is the utilization on each?
There are some differences in what can help your score on Auto scores vs the regulard Fico 8 or 9, but most of what you can do will help all of them.
The first step I would do is reduce the cards that are at the 86% utilization. You had mentioned getting overall utilization down to 29%. That will definitely help, but I would also be sure each card is at or below 29%.
The six inquiries if they coded them correctly should be grouped as one for auto if done within a certain time period. I'm not sure what that is, but others on the forum may be able to help on this.
Glad to see you were accessing FICO scores. Using sites like credit karma are good for watching activity, but the scores are meaningless.
If you want to track your auto score on FICO monthly, you could pay for a subscription either on Experian or on FICO. Experian will give you a monthly update but no activity during the month. FICO will give you changes on FICO 8, but all the other scores like auto when you get a new 3B report. For $29, you can get a new 3b every 3 months. For $10 more, you can get it monthly. I'm getting the monthly 3b because I'm watching my scores closely right now for a future mortgage, but will switch back to once every 3 months after that.
On the 7 cards, are they charging you annual or monthly fees on any of them? You may have acquired these cards before your scores dropped so you may be in a good situation with the cards you have and can grow the credit lines. Depending on what cards you have, this may be a good option to increase your scores.
There are some resources on this forum on steps to rebuild your credit. One of the key steps would be to make sure you have sufficient emergency savings in place, like 2 to 3 months of income. This will help you avoid future lates.
If the goodwill letters aren't working, the only thing else you can do is continue to keep your profile clean and allow the baddies to age off.
You were seeking a new car but said you had 3,000 in equity on your current vehicle? You may want to hold off on the new car until you've achieved other goals, such as emergency savings, and lower utilization on your cards. You might then work on prequalifying, with a soft pull, for any additional auto loans.
As for the auto scores, if you were on Fico.com, I know they will list the negative reason codes to explain what's causing the score. You could look at that to see what the 4 or 5 reason codes are showing to see if there is any additional items to consider to improve that score.
I wish I would have followed this forum earlier in my credit rebuild. So much good advice. Sounds like you're on the right track and just keep digging on this forum and you'll get a wealth of knowledge.
One thing to be sure of is don't pay all your credit cards to zero. You may have already seen that advice. You could lose 20 to 30 points with an All Zero penalty. But you'd get it back as soon as one card posts with a small balance of maybe $10 to $20 (less than $100 is recommended for optimal scoring.
You can also find a lot of information on the score card segmentation for Fico 8. You would be on a dirty scorecard due to the 90 day lates. The scoring is slightly different depending on which score card you are on. My guess is you'd be on a D2 card. Look in the first section for Fico Scoring on this forum and you'll see the posts on Fico8 Scoring.
@coachbeau This can be found easily in two ways.
1) Go to the first section of the forum and look for the Fico 8 Master thread which is usualy on the first page. Scroll down a few pages until you see a Fico 8 Flowchart which shows 12 different scorecards, 8 clean, and 4 dirty. The dirty cards are divided based on public record or derogatory, then based on recency.
2) Look at the link on the top of @Anonymous signature to the scoring primer. Birdman can add more details, but understanding the segmentation and other factors that affect scoring will help.