Washer with sluggish spin

[SIZE=3]Amana Washing machine purchased new in 1991, is functioning normally for all cycles except the spin cycle. During the spin cycle, the machine will make a loud pop-pop-pop-pop sound in rapid succession until the machine is turned off. Also, the water is being pumped from the tub into the drain, however some water appears to be leaking from the vicinity of the pump. The tub spins very slowly (about 5 rpms) during the spin cycle while the motor is. After pulling the front panel off, observed the motor furiously turning the shaft from the transmission during the spin cycle. Also located the source of the loud ‘popping’ noise ….the noise is from a small hard plastic piece that surrounds a splined shaft on the bottom of the transmission housing. This plastic piece rests on top of a matching metal piece and as the spline is rotated during the spin cycle, it appears that the plastic piece is supposed to catch on the matching metal piece. However, but the catch is not made which creates the popping noise. While turning the agitator by hand (power disconnected), the same observation is made of the plastic piece … at a much slower speed. After the two pieces slip, the brake engages for a few revolutions and is then released. A few more turns of the agitator causes this process to be repeated. Belts are not worn or cracked however there is definite wear (black residue powder and somewhat of a glazed surface). [/SIZE]
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[SIZE=3]Also noticed was a puddle of water sitting on top of the pump housing.[/SIZE]
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[SIZE=3]Should I start with belt replacements or does this sound like a transmission/clutch issue? [/SIZE]

It sounds like you have multiple problems, and particular with a leak like you describe and the other issues I think it is time to thank your machine for many years of fine service and replace it.