It seems to me that every boat is a project boat, even new boats right out of the factory
Check out craigslist.
A old Hunter 27 can be bought up here in the NE for about 7 grand, in livable condition.
If you can stand an old 25 footer (less headroom but with a boom tent and an open hatch
it's not that bad, esp in the South), there are lots and lots and lots for about 3-4K, again in livable condition.
If you look around, such boats can be found even cheaper than that, but buyer beware...
Try to find something that the PO was good with maintenance on. Sails and rigging
ideally should be in good shape (if not, take that fact into consideration when negotiating price)
There are some great threads here on finding a cheap, solid boat so I won't go into that
Pretty much all older boats in that price category (and many in a much higher price category) are going to have leaks and such. I bought an old (1981) 25 footer that was in decent shape, but leaked under the rails when it rained. This year I took it out of the water and did, basically, a refit. If you have time but very little money this may be the way to go. It will cost under $1K, including haul out cost, if you are careful (and if you don't have to replace sails/rigging), and will take about a solid week of work.
Take off and re-bed the toe rail, and all other deck hardware. Re-bed and inspect all chainplates. Check under any large cracks in the gelcoat for leaks and rotting or else wet core. Rough up the old bottom paint
, and check for blisters (they will be there). Fill those blisters and re-paint the bottom. Check the rudder for play, and fix if possible. Check all the rigging
, and consider replacing. Check the wiring, if severely corroded consider replacing that too. Service the engine or outboard. (online you can find out how to do all of these things if you are unsure). Check any through-hulls while the boat is out as well. Most likely the original plastic ports will be glazed and leaky, mine had leaked into the core and rotted it for inches around the ports. For $100 get a sheet of tinted lexan and make new screw-on ports.
Buy Don Casey's book, "This Old Boat", used if possible.
Coastal cruiser is the way to go for now, because even an "offshore" boat will likely require lots and lots of mods ($$$) to make it offshore ready. And you don't even know what you will be doing and what you will need at this moment so...
However, very old Pearson Tritons, Cape Dory 25's, and certain Alberg Vegas can be found for under 5K if you really look around, and all are offshore capable with some mods...
Over time you will spend mucho dinero on things you never thought you wanted or needed, and the list of things to do will NEVER be completely done. NEVER. Did I mention, NEVER? Yes, NEVER be completely done, but through this first boat you will learn what is actually important in a boat and the next one will be chosen wisely and with plenty of time and hopefully a more significant budget.