In the U.S., how can you give legal permission for a relative to sign documents for you?

Depends upon what kind of document. VERY few documents require any sort of "power of attorney". You can legally give someone oral permission to sign things for you.  It's only a forgery or fraud if you can prove you did NOT authorize it.  Various legal papers like real estate contracts or deeds and other things within the "statute of frauds" should be signed by a person with a written power of attorney, which was signed by the principle wanting you to sign, and their signature notarized.

You create a Power of Attorney for that person.

You say something like “hey Joe, my hand is still in a cast, would you sign this for me?”  Unless the document has to do with buying or selling land, that’s all it takes.

In order for Joe to prove that you gave permission, he needs witnesses, or a permission paper that you signed. That paper is called a “power of attorney”, and it should say what particular types of documents Joe is allowed to sign with your name.

You have a lawyer draft a limited power of attorney giving another the legal right to represent you in specific matters. And have it notarized.

You sign a "Power of Attorney

".. If you can't find the wording online or buy a standard form, any lawyer can have one typed up for you.

Give them a Temporary Power Of Attorney.