1999 Malibu fuel gauge and A/C trouble

Hi I am new here, I just picked up a 99 Malibu in Immaculate condition for $1,200 with 78k miles. I knew the the last owners it was a old mans car. The car runs and looks great, i do have just 2 problems with it that i was aware of when i bought the car.
The first problem is the fuel gauge, with a full or 1/4 tank or anywhere in between the gauge drops to empty then up then empty. as a guess i would say fuel sender switch in the tank. is this a common problem with this type of car and has anyone had this problem and fixed it? i hate to spend $300.00 on a sender switch if other people had this problem and it was something else.
The Second problem is the A/C is on and blowing cold then just shuts off any idea's ? any help would be great
thanks in advance, Wally. //

First I can't think what it is second on it might need freon or the comperssor is going to give in or the coil needs to be change. Whats does it do and when

Malibus are known for the poor contacts inside the control head- the little green light will be on one minute- then go off the next-- so the fix is to pull apart the control head- and the master switch on the blower motor knob- and wipe clean the contact area- and bend the wiper brushes closer for better contact-- the "grease" that was used turns to hard paste and creates a bad contact area- I use a pencil eraser head to wipe clean the contact area- then some really light oil, not grease to solve the problem permanently...