[Stevebsy]

It seems if AWD works occasionally, then it is probably the solenoid (duty solenoid C) that activates the transfer clutch.

The test is pretty simple, they just hook up a oil pressure gauge to the port on the top of the extention housing. At idle, in D or R, it should be about 7-11 psi, at stall speed ~2700 rpm with the throttle fully open it should be 104-114 psi. With the FWD fuse in, the pressure should be 0 at stall rpm w/ throttle wide open. Plenty of shops could do this testing for you. If you can find a gauge that goes that high and the right adapter, you could do it yourself.
It is outlined in section w3e0 of the service manual, page 121.

The 'thud' noise is the transfer clutch engaging, sending power to the rear wheels in 50/50 split mode

Also...does the power light blink a bunch of times when you start the car? That indicates a problem as well.