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I would say that you have to try those new strings and see if that improves things.
There's not much that can go wrong on a guitar that would affect the tuning of an OPEN string (assuming that you actually get a proper sound through your amp, which it seems you do).
Intonation will affect fretted notes but not open strings. If the strings are old then they won't need stretched. If the strings are not wound on the posts properly you should still be able to get a good strong note on an open string for the tuner.
Besides, anything that needs adjusted can almost certainly be done with the new strings in place.
So - change the strings. Wind them on carefully according to the guides posted here. Make sure you stretch them properly. And finally.... always tune UP to a note, never down. If you go past the note you're tuning to, tune down back past the note then tune up to it again. This keeps the gearing in the tuning pegs tighter and avoids any "backlash" or "slippage".
Good luck.
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