From your description, it still points to a faulty starter solenoid (relay).

You can prove one way or the other by doing this:-
remove the heavy wire between the starter and the solenoid
connect a heavy cable (about 1/4"  diameter) to the starter terminal which
now has no cable attached
firmly press the other end onto battery positive terminal
Be very careful with this cable, wear safety glasses!!

The starter should turn only when the cable is touching the battery positive
terminal

If you haven't got a cable to use, you can improvise by using a normal jump
start lead. remove the alligator clip from one end

If the above is OK, I suggest you put a voltmeter on the out terminal of the
solenoid (the one with the cable disconnected)  it should real nil volts.
Then energise it a few times (turn key to start) with the meter still
connected, to make sure it isn't sticking. (you should get 12 volts)

There are another 2 points to consider
1    The solenoid is not the correct one for this vehicle and works a few
times before burning the contacts together
2    The starter itself is shorting out internally and the extra high
current is burning the solenoid contacts together

Hope this of some help

Trefor