Picked up an Awai NSX-3500 at a yard sale for $5.00 because the lady
said CD player doesn't work. So I though, not bad if I can fix it.

Not a bad deal if you know what to do to fix it.

And the brand is spelled "Aiwa."

I went ahead and got a new laser pickup for it, but that didn't
solve the problem.
Here's what it does:
-Load a CD.
-CD tray spins around to spindle/pickup.
-Laser pickup will move horizontally and laser "eye" moves vertically
trying to adjust or obtain data. During this time, the CD does not
spin even though I've checked the spindle motor and it will run with
voltage applied.

As suggested by another poster, there is a solder short that is deliberately
applied on the pickup which must be undone to allow the pickup to function.
This solder short is designed to prevent static discharge damage to the pickup
during its time in storage and in transit. Usually, the solder glob is located
within proximity of the laser gain potentiometer. Use a soldering iron and
desoldering braid, usually obtainable at RadioShack.

DO take precautions concerning electrostatic discharge and wear a grounding
strap on your wrist connected to good earthing. The internal components of the
pickup can be destroyed by electrostatic discharge.

ONLY remove the solder. DO NOT adjust anything on the pickup as Aiwa pickups
are usually made by Sony, and Sony calibrates their pickups optimally at the
factory.

BTW, your old pickup might still have been good. The CD reading problems with
3 disc Aiwa shelf systems were usually due to excessive dust accumulation on
the optical parts of the pickup. Cleaning the objective lens and airblasting
the insides of the pickup usually brings the player back into operating
condition, although there are no guarantees that the problem won't happen
again, even with new pickups, as this has been a problem with Aiwa shelf
systems with 3-disc changers. - Reinhart