gas gauge reading when full but goes down quickly

Newbie here - so take this as you will.

Usually using an incorrect sending unit will do this. Improper initial resistance (Ohms) can cause this issue if it's not within the correct operating range. Was it an ebay purchase? Is the gauge stock or after market? hopefully someone else with a little more know how can help you out! :)

I'd start by finding out what the gauge resistance is (I hope I'm using the right terms here) and then buying an appropriate sending unit from there that operates in that range of resistance. A quick test with a multimeter can tell you I believe.