The one thing that often gets difficult with these things is that you get charged tax on the signing bonus (as additional income), but when you are required to pay it back you're expected to pay the full amount pre-tax.
Your former employer should take care of the FICA part after you repay them. For federal income tax, you'll claim a tax credit when you file your return according to "Repayment over $3,000" under IRS Pub 525. State income tax may have a similar process, though it can vary depending on the state.
https://www.irs.gov/publications/p525#en_US_2022_publink1000...
In my specific case they sent over an amount to repay that was higher than what I calculated from my original contract, I asked for an itemized breakdown (the original communication just had a single flat number), and I never heard from them again. It was probably more expensive for their lawyers to double check the numbers than the amount they would have received from me.