I own a 2003 Catalina 350. Actual LOA is 36.5 feet. I think the headroom in the salon and forward cabin is 6'7". The model was first introduced in 2002 and there are a number of them for sale on Yachtworld. Asking prices are a bit above the upper end of your budget, but not that much.
Shoal draft on the boat is 4.5 feet. I routinely single-hand my boat and back into my slip when I dock, and I have no problems doing so. The boat has a 35 HP Universal engine and will do nearly hull speed at 2800 rpm.