You will want to review page 27-12 of the Rev. AP G3X Touch Installation Manual.
The trim logic of the G3X Touch system requires that the input pins (11,12) on the GSA 28 servos see +12V on one pin and GND on the other pin for one trim direction, and the reverse of this logic for trim in the other direction. This is done for safety reasons to prevent a simple failure of the trim switching from causing a trim run-away.
According to your wiring diagrams, your installation is simply applying +12V to pin 11 in one direction and applying +12V to pin 12 in the other direction.
If you don't have a GAD 27, you will need to add a simple relay deck (which provides separate pitch/roll control) that is actuated by your stick grip switches to create the required drive logic with your existing stick grip.
Update: Looking at the Ray Allen wiring diagram for the G207 stick grip, it seems possible to use that stick grip without a relay deck. You should probably re-check your wiring against the RA diagram.
If properly wired, there is no reason that applying servo power would cause the fuse providing power to the stick grip to blow. The stick grip shouldn't even be outputting a power signal until one of the trim switches is pressed. When servo power is applied, the bypass relay inside the servo opens, and the trim inputs on pins 11,12 are no longer directly connected to the trim motor outputs on pins 13,14.
I wouldn't increase the size of the stick grip fuse above what the manufacturer recommends. This could damage the stick grip and perhaps other system components.
I would do some basic trouble shooting with a meter and the power off to find the wiring path that is shorted to ground. You can also unplug both servos, and plug them back in one at a time to further isolate the wiring problem. According to your wiring diagrams, you have an unusually large number of connectors in your system, which adds to the complexity.