Fuel gauge shimmy

I think I know the answer to this but before I tear it apart just want to make sure..

The fuel gauge shimmys when I'm driving..Is this a tank sending unit problem?

Prolly not. More likely the resistor on backside of the gauge panel for the fuel gauge. Then again, being a '77, it might not have the resistors on the gauges like earlier cars had. Not sure..

I have had this problem a couple of times over the years on my 82CE.  It always came down to a faulty sending unit in the fuel tank.   I never had a problem where the fuel sending unit registered empty or full all the time,  my problem was more like yours, the fuel gauge needle would bounce around.  Both times the little metal arm that slides up/down the wire coil to measure fuel level was making a lousy contact.  I tried cleaning with limited success (worked for a few months), so now I just replace when it goes bad (due to the effort and contortions to get that dang thing out).  I know my 82 does not have that resistor on the back of the fuel gauge.  As Joel says, if you have that resistor, that would also be my first suspicion. 

My 77 has the resistor it should read 90 ohms

I had a similar problem with my 1970. I replaced the gauge. No problem now. The only problem is getting to the gauge to replace it.

I've got the gauge just trying to get access to it to replace. My hardest problem is getting the aftermarket radio out from the gauge panel..I need some sort of "key" I was told so the radio will release from the cradle it sits in..The radio is 20 years old and a Pioneer.

Can't help with that one. Have you tried your ignition key to unlock the radio?