Kiira,...
I recently had my 238 repaired by Tascam Factory Service, in Montebello, CA.
It was doing the exact same thing as yours,... winding up and whining like a 747. The tape speed was going full-speed-ahead, with no speed control whatsoever. WARP SPEED.
It cost $239 to get it fixed. The motor was replaced. The price included the motor, labor, and the "shuttle" knob was replaced, as it was missing, so deduct a small amount for that, in your case.
That was, if I can remember correctly, $125 for an hour of Tascam labor, and the rest of the amount was the MOTOR and the knob. It was not cheap, but it was very much worth it.
MY analysis of it, is that it was better to get the 238 fixed, as the used price for a 238 on Ebay is minimum of $200 + shipping, or even higher for the mint or v-clean 238 unit, up to $350, or more, depending on the day. AND, you never really know what kind of unit you'll be getting off Ebay. F/I, the 238 I got, was "supposed to be working fine, but untested", and I thought I got a great deal for $200,... however, it didn't work & had this motor problem, right out of the box, & in the end cost me the above mentioned $239 to fix.
Word to the wise,... is not necessary. Heh.
For the rest of us common folk,... I'd say by my experience, of REAL WORLD EBAY "DEALS" and REAL WORLD REPAIR BILLS, it is more worthwhile sending your 238 off to Tascam for authentic Tascam Factory Service. You'll at least be knowing what you're getting, the history of the unit, and getting the utmost in quality repair, by Tascam itself.
That's my take on it. You may get a different opinion, for however many people you ask. Take my advice, if you want.