You can get your FICO8 scores and reports by signing up for a $1 trial at CreditCheck Total, then canceling. You can do that several times if you wish before you decide to pull all of your scores from myFICO.
If you list your cards, APRs, limits, and balances, people here can help you with a paydown plan. Given the upcoming mortgage, I'd assume that scoring is a priority over total cost right now.
Also, if your mortgage is coming up in the next few months, don't apply for any new credit until the mortgage has closed and you have the keys. 
The generally accepted FICO thresholds are at 8.9%, 28.9%, 48.9%, 68.9%, and 88.9%. They're something you can keep in mind as you pay things down. These apply both to overall utilization and individual cards. 8.9% maximizes your score. 28.9% is considered responsible borrowing. 88.9% is maxed.
Getting below 88.9% and 8.9% are almost certain to help your scores. Crossing some of the thresholds in the middle will probably help too. But bank on only some of them helping your score. Which ones would depend on your profile, and the ones that would be helpful are hard to predict.
Another scoring factor is the number/percentage of cards reporting non-zero balances. To start with, you'd want non-zero balances on less than half of your cards. One card with a non-zero balance is ideal. You don't want all cards with zero balances as that will ding your score.
From a scoring and financial (interest) standpoint, you're definitely best off attacking your cards rather than the loan. That's going to save you money on card interest, and it will bump your scores and hopefully land you better a mortgage interest rate.
From a DTI standpoint, paying down the car loan — without paying it off — would be more useful. That's because when computing DTI, minimum credit card payments are used rather than full balances. I'd submit that your car loan is probably in a good spot; we rarely hear about reasonable car loans getting in the way of mortgages. The reason for not paying the car loan fully off is that scores are dinged for no open installment loans.
There are two things we don't know: when a mortgage might occur and how fast you'll be able to attack your debt. Maybe you can give us an idea. Without that information, I'd suggest attacking the two cards with the highest APR first. Because the balances aren't among your highest, that would also have you snowballing a couple of cards. If you smell the mortgage coming on, you can shift gears toward the high-utilization cards.
You might want to consider a balance transfer from one of your high APR cards to your new Discover card. I'd try to keep utilization on the new card at 48.9% of its limit or below.
After you've had the Discover card for a couple of months, check in periodically for the availablity of limit increases. They'll be soft pulls and won't count as seeking new credit before applying for a mortgage. Discover has no rhyme or reason when it comes to approving or denying CLIs, and you don't reset any timetables by making requests. If there were a timetable, I'd suggest holding off on CLI requests until your card debt had markedly improved.
Someone else is likely to chime in with a different plan of attack. That's good as it'll give you a smorgasbord of ideas to choose from.