You're working on one end (raise income).

Try working on the other end (reduce expenses).

Write out your budget.

Write out your expenses.

Decide what can be cut.

Decide what can be reduced.

Decide what is necessary.

Reduce your balance, but since you are asking about this, then that's not feasible obviously. All you can do is ask them, but since the minimums are done by a predetermined formula, it's highly unlikely. You are better off asking for a reduced rate. At least that will slow the bleeding a bit.

Are you (or your partner) able to get a credit card with a 0% APR transfer option and transfer at least some of your current balance out? Some cards are offering up to 21 months of 0% APR.

In general, your card's required minimum payment will simply be a fixed percentage of your owed balance. Usually at a level just above the card's APY divided by 12. That'd be enough to cover that month's interest charge plus a (small) amount left over to pay down the balance. There's usually a "floor" of $25-35 that the required payment always stays above, but with a $10k balance you're probably a ways from that mattering.

I'd expect the required minimum to be set in the card agreement and be non-negotiable. But it might be worth calling the card and asking, you might get lucky. Though be aware that even if they allow that, if your payment is lower than your monthly interest charge then your balance will increase month to month, even with no new charges.

Honestly, the only real way out is what you appear to already be doing: get income larger than spending to have more money available to pay down the card balance faster.

But a few potential short-term bandaids:

• Do you have the ability to make any more cuts to spending? It sucks, but if you focus on it being only until the card is out of your life, that can make it easier. "Eat out less", "make coffee at home" are cliches, but can help, if only a little. But every little bit helps.
• See if you can be approved for 0% transfer card. Moving this $10k over to that would remove the interest you're paying on this balance each month. Those transfers do tend to have a fee, usually a percentage of the balance. So you may have to pay $300-400 to get that lower rate, and it'll only last 12-18 months.
• Borrow money from elsewhere (personal loan, friend, family) to zero out this card and pay that other loan back at a lower rate.

You can try to see about getting a debt consolidation loan. Here is a website showing ranges of rates and amounts you could potentially get. This can be good or bad depending on what you qualify for. Even if you're stuck at something like a 4 year term with 20% interest you'd pay $304.30 monthly, which is about the same as now. The difference is, it will be paid off in 4 years, whereas your current cards could take much longer. If you score and manage to land something like a 7 year term with 10%, you'd be paying $166 a month.

You can also look into balance transfer credit cards. These are promotional cards, with 0% interest for a limited time, usually around 1-2 years. This will help pay down your principal during this time, so it'll reduce your monthly minimums. The caveat with these are that they will likely have balance transfer fees around 3-5%. That means you'd be in debt $10300-$10500 if you transferred right now. Not great, but that's still better than the 20+% you're likely dealing with.

In general though, you should just get a full-time job in order to pay it back.

Can you sell stuff to pay down the balances? You are in emergency mode: dump the gaming consoles and eating out and ANYTHING you don't NEED to live on. eBay, Craigslist, local Nextdoor postings -- sell everything you don't need.

Pride's got nothing to do with it. You should be proud you are facing this stuff head on -- take that win. And do EVERYTHING you can to show your partner your appreciation for their efforts.