GFCI breaker with GFCI outlets trip with no load
This is probably a leakage problem

GFCIs are differential trips -- if too much current (more than 4-6mA) is going out but never getting back to the GFCI (remember that current flows in loops!), they trip on the assumption that current is going somewhere bad (like through you).  As thus, a variety of leakage sources can trip them:


Cable leakage due to water in non-wet-rated cables, degraded insulation, or physical damage
Faulty or intentionally leaky appliances or fixtures
Capacitive leakage on long lines
Water in splices and boxes causing leakage currents to flow
Miswiring that routes current around the GFCI


In your case, even with 350' of cable, you should be fine -- you'd need a cable running at 200-300pF/foot of capacitance to get into trip territory from capacitive leakage alone.  (I can't find any figures for UF, and only a passing mention online of 20-30pF/foot for NM, nor do I have a capacitance meter and cable samples otherwise I'd test it myself.)

So, once you rule out a miswire, I'd have your nearest friendly electrician megger the cable hot-ground and neutral-ground to make sure the insulation is good -- make sure to disconnect the cable at both ends when doing so though!  If the buried cable passes the insulation test with flying colors, then I'd check the devices at the load end for water infiltration or insulation damage.