Install a CD player?
Seriously, though, the tape switches when the deck thinks the end of the
tape has been reached. There are a variety of ways this could be
detected, but my guess is that the drive belt(s) are worn to the point
where the tape does not move, or move fast enough, and the deck thinks
you're at EOT and switches direction. The belts are easy to change and
probably fairly inexpensive if available, but you will have to remove
the unit and take it apart to reach them.
If replacement belts are not available (which would not surprise me 23+
years later for a technology that has largely fallen out of use), it is
very likely but by no means certain that O-rings would make reasonable
replacements. It might take some searching to find someone who carries
the diameter and sizes you need but keep looking--O-rings are available
in a wide variety. The lifetime of O-rings used as belts is probably
not close to the lifetime of the original belts, but I would expect them
to last a while, at least, and you could try different materials because
O-rings are made from several different materials including BUNA-N,
silicon, and Viton. I'd try them in approximately that order because
BUNA-N's are dirt cheap.
Good luck,
I had exactly this problem with the factory cassette player on my old
'86 535i and the fix was exactly as JRE describes, although I was
lucky and at the time I was living close to an electronics supply
place so I was able to buy the exact correct drive belt for it.
another issue I had was that I had to buy a more expensive cassette
adapter so I could listen to my Discman in the car (remember those?
dang I feel old now) because the cheap ones that used to be sold would
not rotate the non-drive spindle and therefore the deck would think
that the tape had run out and keep flipping sides.
Today, I would probably recommend replacing the head unit because
there's so much cool stuff you can get that wasn't available in 1995.
I just replaced the head unit in my F-150 because it was a radio only,
I got a unit with HD radio and the sound quality is amazing. Of
course I live near a major city so most of the stations come in in
digital, might be of less use if you live farther away. Satellite
radio is another option that I'll probably look into on my next car
(current company car does not have an "aux in" jack so I didn't
bother, but I'm only a couple months away from ordering a new one.)
Cost me less than $200 from crutchfield (probably could be had for
cheaper if you shop around) and install took less than an hour,
including soldering the adapter harness to the new head unit's
harness.
nate