As the link in your OP says, in the USA you do not have a legal signature. So the question for you is what signature do you want to change? That will give the answer to how you can change your signature,
Your signature is used as a evidence that you signed a given document. If you said "no I di not" then potentially it could go to court, and the side claiming that you did sign might produce other evidence, or have a handwriting expert examine what you say is your signature, etc.
- For really important documents, often the parties will want a notarized signature. This means that someone who has a commission attests 'Yes, I saw her sign this."
- For something somewhat important, they might have a witness, but not a notary.
- In some cases, they might have a signature on file that is then compared. So the bank has a signature card (or now electronic on a pad). They can then compare a check to that signature. So to change your "legal signature: for checks, go to the bank and sign a new signature card.
- For something like mail-in voting, they check against the signature you used when you registered. Maybe that is the signature on your driver's license. Honestly, I think you would have to go and get a new driver's license. If it is not time critical (like you want it changed for this election), you can just use your new signature next time you come up for renewal.
- Speaking of driver's license, that is used to check your signature, is in a way it is the closest we have to a de facto "legal signature," in the same way that the card (or a non-driver ID card from the same place) is our de facto "official ID." So again, go get a new driver's license or just use your new signature next time you need a renewal.
- Go to the bank (all of the banks you have an account with) and sign a new signature card.
- Go to the DMV and get a new license, or just change your signature when you need a renewal.