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I also had those experiences, and found out that GFCI was NEVER the fault.
They fill a function and are not "overly sensitive". Here in 230V/50Hz land GFCI is required to trip at 30mA earth current in order to be classed as "personal protection". Choosing the higher current rating will certainly make it less sensitive but also worthless as protection device as more than 30mA through body will likely kill you.

It's quite easy to troubleshoot. Break the circuit, remove the breaker and measure leads going FROM GFCI with ohm-meter. Resistance between earth, phase and null should be very high, almost infinite. If resistance in-between any of those leads is less than that, you either have a short (>10 Ohm) or a sneaky-leaker (10 - 100kOhm) due to water, a nail that went through the wire or something like that. First one is easy to fix, sneaky-leaker is dangerous one.

Common fault scenario is short between earth and null (or whatever you call it in 110V-istan) in outlet itself. It will not burn the fuse (as earth and null are shorted in the fuse box anyway) but it will trip the GFCI. Next most common scenario is "in between, not-quite the short" (the dangerous one), as it might electrocute you or start to burn without burning out the fuse. And this scenario is the one GFCI is there to protect you from.