First off, welcome to DIY; it's not as hard as you think!
Next, room pressurization occurs in all rooms, it's just more audible in small rooms. What's your short dimension? (565 / distance in feet) is formula for the lowest frequency mode that fits. Granted, the effect is greatest at frequencies below the lowest supported room mode, but that's a 56.5' room at 10Hz.
As a starter sub, sealed is the easiest and generally smallest. It's hard to beat a combination like this:
Driver: Dayton RSS390HF http://www.parts-express.com/pe/showdetl.cfm?partnumber=295-468
Amp: Dayton SPA500 http://www.parts-express.com/pe/showdetl.cfm?partnumber=300-807
or Yung SD500-6 http://www.parts-express.com/pe/showdetl.cfm?partnumber=301-514
(mounting hole already cut to fit RSS-390 driver.)
This driver/box combination gives an F3 of 35Hz and an F10 below 20Hz, with the ability to output 100dB (@1m) at 20Hz from 250W. That sounds like what you're looking for. I suggest the SPA500 amp for it's PEQ, and the Yung for it's price ($200) and 6dB of boost at 25Hz. There are also rack-mount options that may fit better if your system is rack mounted.
It also hits your cost target; $175 driver + $200 amp + $100 box <$500. Yes, there are better drivers, cheaper drivers, larger/smaller drivers and plenty of combinations that may work just as well for you; this is just one of many good designs, but it is one of the good ones.
Since DIYSound is out of the 3 cu ft packs, I'll note that if you can find a "hobbyist" or "fine hardwoods" lumber store (i.e. not construction lumber), they may be willing to cut the panels for you. HD/Lowes may, too, but dimensions will not be as precise... and you'll need some basic tools like a drill, and a jig saw for cutting mounting holes (and for the driver cut-out unless you have a router and circle jig).
... and I now see others have responded similarly! Enjoy!