When you click on links to various merchants on this site and make a purchase, this can result in this site earning a commission. Affiliate programs and affiliations include, but are not limited to, the eBay Partner Network.
Sending unit is easy to check with an ohm meter. But the swinging sounds like a bad guage.
The connector for the sending unit is in the left hand side quarter panel. Unplug it. Connect your ohm meter across it. Have your helper raise and lower the float arm using a bent coat-hanger through the filler hole. It's a zero to 90 ohm sender. Pretty standard fare.
...one of the easiest & cheapest things to do for this is to check and clean the black ground wire going to the frame back in the left rear corner, at least on a '68. This fixed my swinging gas gauge instantly.
Most likely bad electrical connection (corrosion) on terminals under rear of car. Check the ground connection first. Remove, de-rust, wipe clean, reassemble and test the result. If that fixes it, coat the connection with sealant that will keep water/air out.
I had this problem 2 years ago, shortly after buying my 2014 Z51. Techron treatment did the trick.....for a while. When it failed again Techron didn't work. The right fuel tank gauge failed. My extended warranty paid the $3000 bill to put in the new sensor which was mechanically stuck. This past November it happened again. This time the left side sensor failed and the left side fuel pump had a low pressure signal. Once again my warranty paid the $3200 bill. So your fix may be costly. Since most of the cost is labor to drop the whole rear sub-assembly, change both sides at once! Good luck.
Ken
I filled my gas tank and the needle shows wayyy past full. Drove for 1/2 hour and needle swings between 1/2 tank and full tank. Bad sending unit??
Here's a video from Willcox Corvette where they do testing for different fuel gauge operation scenarios. Maybe this will point you in the right direction?