The credit union may want to see the vehicle and take pictures before giving you the check. I've seen it both ways. Sometimes they just say bring it by after so they can take a picture. If you negotiate a good price, the credit union will still give you the full amount of the value they've approved for the vehicle, and you can pocket the difference if you want. Basically, a cash loan at a used car loan interest rate. Not bad!
For registration I believe you only need the signed title. You can take a bill of sale with your purchase price but the state may just charge you based on their valuation of the vehicle. The owner should keep their plates, and don't think your bill of sale will let you drive it for a week before registration because the police will ticket you. I believe you have only 24 hours, at least that's what one cop said. It may vary depending on who you talk to.