Start with the variables that can be eliminated. Get new strings making sure to lube their sliding points of contact (saddles & nut), make sure they're stretched to the point where their pitch doesn't change after doing bends or dives, then intonate the guitar properly.
Once that's out of the way, then look at your technique. If notes are still going sharp, determine if it's due to you either pressing too hard, or possibly even pushing them slightly as you fret the notes. If it is, you can try 2 things: Either work on fretting the notes straight down, with only as much pressure as needed to make the note ring clearly, or if it turns out to be a habit you can't fix, you could intonate individual strings slightly flat, to compensate.
You could also go up in string gauge to see if that helps. Theoretically it should help counteract you fretting too hard. How much it would help is hard to tell without just trying a heavier set.
But overall, if it's you causing the problem, you'd be best served by breaking that habit. Someone posted an exercise in one of my Theory & Technique threads, in which you play through whatever you're currently playing, but only placing your fingers lightly on the tops of the strings. Iow, not even fretting them. Suppose to help you develop a lighter touch. Which is not only necessary for not making your notes go sharp, but also helpful in playing faster.
I feel it's safe to say having the lightest touch necessary to fret the notes, and nothing more, is the best fretting technique for any guitarist. But once you get that, you'd probably want to keep the same string tension across all your guitars, to keep the feedback your fingers get, consistent. Which doesn't necessarily mean all of them having the exact same string gauge, due to various scale lengths.
Iow, having 10's on a LP, and 9.5 on a Strat keeps that string's tension closer, than using 10's on both. But that's just for the high e. You'd have to run the numbers through a string gauge calculator to see how the other strings' gauge/tension shake out.
