Fuel Gauge and Temp Gauge stopped working while running:


I have a 1992 Caprice Classic and I changed the battery 3 days ago. Drove it for one day, everything was fine. Car was parked the whole next day. Took it out today and on the way to work I noticed that the fuel gauge had shot all the way past full to the point I couldn't even see it (was 75% full as it showed when starting in the morning). Also, the Temp gauge (which usually sits at 25%) had moved up only about half of that and then was stuck.
I let the car cool off for about 4 hours. The gauges are still where I saw them stuck. Car still starts OK and when the key is turned to ON (just before ignition) the speedometer needle shows a very slight movement (which is normal as before).
Checked the fuses on the internal box (near the door, under the dash) and all are OK.
I know something like this has been addressed in some other forum/thread but would really really appreciate fresh answer/suggestions with little easier to understand terms.

 

Didn't notice something till going back tonight. Beeping sound comes on as soon as headlights are switched on. Headlights working, its just the annoying beeping sound which won't stop till I turn the lights off.

Looking forward to some help ...