The 92-93 mini starter has to be wired different than the 1991 and older model big starter to make it work right - there are 2 wires going to the mini starter - the little one and the big one -- up at the solenoid you take the big red wire off and put it on the battery side of the terminal and put the skinny wire on the switched side of the soilinoid and that will fix that problem for good - I do not know if the 1992 and 1993 solenoid is different than the 1991 and older ones or not - check that to make sure. You need to know that the mini starter has a built in switch that makes it work that way - I know it sounds strange but it is true. Been there - done that.
I would also note that using two solenoids can cause the starter to stay engaged longer than it should. I would have run the skinny wire right to the new starter's piggyback solenoid.
My stock mini starter did the same thing at 130,000 miles. It worked fine, but after it started and the gear retracted back in the housing, it was still spinning and because the bearings were worn, it made a squeely sound. Very embarrasing! I just bought a new one. No problems.