Any socket that will fit and you can get securely on the truss rod nut will work, but don't tighten it up too much. No more than a half-turn at most. If it's very hard to tighten, don't force it. You can strip the threads or even snap the truss rod.

When in doubt, take it to a good luthier or guitar repairman/tech. It's easy for them to tell what needs to be done and a set-up like that is pretty inexpensive. Far more inexpensive than having a truss rod replaced.