(assuming you are in the USA in a place with laws similar to where I live)
Yes, you'll be fine, as long as both names belonged to you at some point. Endorse it for the name it was written out to (sign your old name). Then, under that name, sign your new name. Some people won't take a check like that because of risk of it being fraud, but your bank almost certainly will, particularly if you deposit it into your account there. They probably won't ask why (my wife deposits checks made out to me, after I sign them, and they never ask why - I just sign them, then she signs under my name), but if they do, just tell them that both names belonged to you and you have changed your name - if you have legal proof of that that you can bring, all the better (you probably won't need to show it). If they won't let you cash it, ask them what you should do.
The bank will likely have no problems if you deposit it (not cash it) because you've already shown them who they are - they know if it was fraud, it's a pretty stupid fraud because you're going to get caught. So they won't worry about it, unless you think your family will charge you with fraud (if that's the case, I probably would just return the check saying you can't cash it).
It's no different than someone who changed their name when they got married.
Note that they may place a hold on the funds for a few days while it clears, perhaps longer than would be typical if the check was in one name. So don't spend the money until you know they've released the funds to you (you can ask how long that will take when you deposit it - just ask, "When will the funds for this deposit be available?"). And all bets are off if you try to cash the check - they'll be a lot more careful if you try that, and be a lot less willing to take risks.
If you don't want to risk dealing face-to-face with someone who might not react well to your name changing from one gender to another, an alternative would be to drop it in the night deposit box after-hours with the check endorsed as I mentioned, in an envelope with a filled-out deposit slip. They will call you and/or mail it back to you if they don't like it.
You also have my sympathy for dealing with a lousy family. You might "suggest" to them that next time they just leave the "Payee" line blank and let you fill it in, since they don't seem to be able to remember your legal name.
