capable world cruisers under $10k
As already mentioned, I think your biggest challenge is not just finding a suitable boat in your price range but eventually getting to Blue Water aboard a seaworthy boat for $10K. And regretably, the comments about a ''hot shower'' are reasonable only if you use the cockpit, at least aboard boats you''ll likely end up seriously considering.
As one example, look at an Albin Vega. They have circumnavigated, done many, many trans-Atlantics, were well engineered and carefully built but to a very basic level of fitout, are a delight to sail (and can be made to sail using sheet-to-tiller methods, which eliminates the cost of an autopilot and/or windvane), and some will still have inboard engines.
Assuming you find one below your $10K threshhold, won''t you want to replace the (most likely, quite old) rigging? What work will the engine require? Nothing is more ''basic'' than the sails: what servicing and addition(s) to the inventory will she require? And then there''s good anchor gear and a decent anchor platform on which to hang it, late 20th century nav lights that can actually be seen, a radar reflector, and a decent stove to cook on.
It is probably possible to make the numbers work but it will be challenging to find the right boat with enough of the essential pieces of gear. The critical issue is finding a low-cost boat that''s still seaworthy enough - in design, construction and current condition - to be up to the task.