I need a musical sub under 500,,,,new or used.

VMPS New larger subwoofer..it sells for 700$..I don't think you can find a used one..no one sells them on the used market..why?/because they are so good.....

The Aperionaudio 12 recently had quite a good review.

Hsu distributed by Ascend Acoustics or Parts Express sells a sub. These would be two options.

Lots of suggestions so far. I'd just add, at this price level, avoid anything with a woofer more than 10" in diameter. The amps in the cheap subs and even in some more expensive ones (like your 15" Velodyne) are not capable of controlling that woofer cone. They are "slow" and sloppy, as you have heard.
Find a sub with a 10" or two 8" woofers made by a company that also makes good speakers for music.
You didn't say what speakers you're using, so I can't say where to set the crossover. But, from what I've seen, a lot of people have their subs set way too high in the frequency range, over-lapping the performance of their main speakers, and that's why they sound so bad.