That is a really tough order. In order to buy a 30 to 35 footer under $20,000 you are probably talking about a boat that was constructed in the 1960''s or 1970''s. Many of the boats of that era were built with a lot of fiberglass but not particularly good engineering or sound building practices. So in theory you are looking for a better built boat of that period.
It should not be hard to find a better constructed boat in your price range that has a good hull that is capable of making a transatlantic passage. The bigger issue will be all of the parts attached to that hull. When you talk about doing a transatlantic passage in a 20 to 40 year old boat, items like chainplates, deck and mast hardware, standing and running rigging, wiring and plumbing, rudders, sails etc, are likely to be past the point where they are safe to be taken offshore. Engines, batteries, electronic systems, head and galley gear are likely to need rebuild, upgrading, or replacement. Keel and structural bulkhead attachments would be suspect. And so on......
In theory boats like the Pearson Vanguard, 323 and Alberg 35, Allied Seawind 30, Luders 33 and Seabreeze 35, Bristol 29 and 33, Chesapeake 32, Cheoy Lee Bermuda 30 and Offshore 31, H-28''s, LeCompte Medalist, Tartan 30 and 34, American Marine Galaxy 32, C&C Corvette, Seafarer (1960''s era Rhodes designed 33 footer whose model name escapes me at the moment) and so on are designs that should be capable of an Atlantic crossing but at the price range that you are proposing are likely to be too tired to be trustworthy and putting one in shape could easily cost as much as the purchase price of the boat. The key would be to either think smaller, perhaps 27 to 28 feet, or to try to find that one needle in a haystack boat that some one else rebuilt for that purpose and then ran out of steam.