During the past several years, I looked for and did purchase three sailboats for well under $2000. I purchased these all from eBay auctions (two were charity auctions). I did need to be patient and did my homework for the particular models. I am happy with my choices and do enjoy the work involved.
Boat #1 - 1983 O'Day 19 with trailer and outboard for $1600
This one needed to be cleaned, registered, then launched. It is very easy to rig, has a swing keel and can use a standard power boat trailer. I am keeping it for my thin-water sails. I like its simplicity for day sailing.
Boat #2 - 1988 Catalina Capri 22 wing keel and outboard for $1200 (no trailer)
Again, this one needed only to be cleaned, registered, then launched. I chose to remove the flaking antifouling bottom paint since I planned to trailer-sail. But without a trailer adjusted for the wing keel, I did not get it into the water last summer. It is ready to go into the water now.
Boat #3 - 1978 Cape Dory 25 for $1200
My wife liked the woodworking on this and told me to bid on it. Yes, sometimes it is important to obey ones wife! She liked the extra length, full keel, and mass (4000 pound weight) for going into the open water.
This boat did need much more work. The hull did have a serious case of blistering (after some serious sanding sessions, it is now ready for the sealer, then paint) and all of the hatches, portals, etc. were removed. Backstay was kinked and needed replacing. When the work is all done, the boat should be worth more than what we will initially put into it (note the word "initially").
Only you can answer the questions needed to be asked - HOW DO YOU INTEND TO USE THE BOAT? And what are you capable of doing for yourself?
Most people are really day sailors. Our O'Day 19 does have a cuddy cabin and can be used for weekending and "camping" - plus have some privacy for a portable toilet (required by at least one of us).
Trailering keeps costs down.
The O'Day (12" draft swing keel up, 1500 pounds) is so easy to trailer and setup and launch at most any ramp. I don't hesitate to go on a moments notice.
The Capri will need a deeper ramp (2' 8" draft, 2250 pounds) but I can raise the mast by myself without a gin pole or a-frame. Rigging takes a bit longer than the O'Day but I have not practiced on this boat as much.
The Cape Dory (3' draft, 4000 pounds) "can" be trailered, but would need a mechanical advantage or a gin pole or a-frame to raise the 100 pound mast. We have a slip for this year.
My choices fit my situation. Next year, I plan to have at least one boat less. We are still examining what works for us. It is possible to get a boat within your price range, but you may not get everything that you think you want in the first purchase.
John