Battery draining

95 Jeep Wrangler, 110K miles, 4 cyl engine. Battery was drained in the mornings. I replaced battery and and tested the old battery. Battery was at the end of service life but still not dead. From the negative battery post of the new battery I connected a test light to the negative cable and got a bright light. I measured 12.74 V and 1.25 amps with my volt meter.
I removed and checked 1 by 1 all fuses that I could fine under the left side of the dash but the test light stayed on. I took the removable face of the radio off and I removed the cigar lighter-same result. What else can it do?
Side note: not sure if this is important. But before the fuel gauge would always be inactive without ignition key turned on. Now it shows the fuel level even with the negative battery cable removed.