I wouldnt buy 1960-anything,especially if you want to stay true to your budget.
I bought my first boat 8 years ago for 5k.ready to sail.
It was a 1975 mirage 24.They were designed by c&c in the early 70's,
originally planned for a one design class that was aborted due to another 24 design with a higher freeboard and their 25 was only a foot longer with way more cabin space.
Mirage yachts (c&c dealer in quebec) bought the plans and built lots of them (this is the boat that started mirage yachts in 1974).
Awesome light wind boat,sails and points extremely well,solid construction,standing headroom is about 5 1/4 feet,it could sleep 4(3 comfortably),had a head with holding tank,sink,stove,hanging locker,fair amount of storage space for its size,nice sized cockpit with 2 large sail lockers and a sloped deck.
I put mine thru many storms and high winds and always felt safe.
When I bought mine it came with a bulletproof honda 9.9,new harken furling,new winches,lines,blocks,cleats,traveller,cockpit cushions, 6 sails all in perfect shape and 1 spinnaker.The only issue it had was the nasty plaid interior cushions which I replaced for $1000 and leaky window seals which cost $70.
There are a few other boats I can reccomend but you remind me of my own situation when I bought this boat.
I was 22, had around 6k and just wanted to sail a boat I felt safe on that could actually sail well.
I couldnt find anything but crap for that money and noticed most people were unloading these for next to nothing due to bieng a non trailerable boat with a lack of headroom and bieng too old for racers, however, of the 4 or 5 I looked at I noticed all the boats had solid hulls and decks which is atributed to their build quality and I knew they sailed well.
This pic was taken from my dinghy with my freind sailing her.