Can I somehow make my ex sign his rights away to my son so that my fiancé can adopt him? His biological father hardly ever sees him and my fiancé raises him...

No, you can't "make" him do anything. You can only ask.

Your title is incorrect. Your fiancé is not your spouse yet.

You will also need to wait until you are married for your fiancé to adopt him. Other wise you will need to sign your rights too.