No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Either call synch or dispute the account with the CRAs as not yours. Synch is a better first option though.
I'm not sure about Synchrony specifically, but I know in general an AU account including the balance will stay on your credit report until the next statement cuts and the company updates. Even after it is updated as 'relationship terminated', the account will still show on your credit report, just with a balance of $0.
If you want it removed completely, I recommend calling the credit bureaus directly. I literally had this situation with one of my husband's cards on which I was an AU. I called all three Monday (5/4) and I got an update today (5/6) that Transunion has already removed the account. I suspect the others will follow shortly.