Moved to On Topic.
There isn't enough here to make this relevant to the Integra forum.
So you want RWD for under $6K, with decent handling and the ability take significant increases in power. You can rule out any WRX or EVO at that budget. Frankly, that budget is a killer.
h:
With California prices, $6K will get you a decent 94-97 Nissan 240SX. But you'll need to spend anywhere between $3K and 5K to either turbocharge the KA24DE motor or swap in the SR20DET from it's Japanese cousin, the Silvia.
Another option would be a late model Mustang, but again with $6K as your cap, you're limiting yourself to high-mileage fix-er-uppers. A shop-worn Mustang will eat another $3K in repairs easily.
Another thing to consider is what auto-x class you want to compete in and what limits that will place on your modifications. You can have plenty of fun in the slower run groups, but then all your aesthetic concerns come into play.
I have no idea what cars you consider ugly.
Would a 96-01 Civic pass your test?
Or a early 90s Prelude?