I'm getting a fake short on my fuse tripping GFCI
Gfci’s trip at aprox  5 ma your fuse is 15 or 20 amp so this small leakage would not be enough to blow a fuse or trip a breaker and the issue could have been there for years.
1950 wiring is usually cloth wrapped and the insulation can be in bad shape.
Measuring resistance on a plug in device will not tell you if it is the problem many devices that have motors or transformers will show close to a short across the blades that plug i. this is a dc value, ac values are many times this reading.
The leakage also has to be to ground or not running through the hot and neutral. Once the imbalance is ~5ma the GFCI trips and is referred to as a ground fault because some place in the circuit current is leaking or shorting to ground.
How to trouble shoot a ground fault with nothing plugged in, first you could remove the wires off the “load” terminals this will verify the GFCI is working when you press test and reset. If that works re connect the load to the GFCI.
the next step count how many outlets are fed by the GFCI (any outside lights?)
Cut the number in half and with the power off inspect the wiring at that receptacle. If you don’t find any problems some times I will pull the hot and cap it then see if the circuit will hold. If the circuit trips we start working back towards the GFCI if it holds we work towards the end of the run each time cutting the total in half and repeating the inspection until the fault is found.
If no fault is found again verify outside lighting or receptacles, a little water in a outside receptacle won’t trip a fuse or breaker but will trip a GFCI.
I usually narrow it down to the segment of wire or receptacle in 4 cycles by using this 1/2 cut rule and working to the fault. After verifying that the GFCI holds without a load.
Remember anything on the circuit can cause the fault your lights could be fed for the room or outside as I mentioned. If you see more wires in a box than just the feed from the last device and the cable going to the next run that could indicate a branch in your circuit to another location and outside fixtures have been the culprit many times for me.