Likely not a cause of the loose halyards, but replacing them is a good overwinter project. The size needed is as much a matter of hand grip as stretch... 5/16” is too small, 3/8” feels better.
You can banjo string the main halyard. Hoist, put a turn on the cleat and pull the halyard aft to raise it some more. Take the slack in on the cleat and repeat.
And yes, take apart and inspect the shroud turnbuckles. I have seen older ones collapse and break without appearing weak. There should be no corrosion or dirt built up inside, and the threads not worn. Replace as the wallet allows.
I added a winch handle holder to the mast and parked a cheap handle under the boom. Easy to use and stow for hoisting.
https://www.defender.com/product.jsp?id=1611093
