One thought about that note you're hearing. There's a thing called "sympathetic vibration", which means that when you pluck one string, all the others will try and resonate as well. Usually this isn't too bad, but if there's a harmonic on another string which matches the note you're playing, the other string will resonate strongly. This reaches its upper limit with things like mandolins where there are two strings tuned to the same note.
So what you're hearing could be one of the other strings resonating. This could be what's making you think the note is changing - the harmonic on the other string will die away quicker. Try damping the other 5 strings with your left hand while you tune each one, and see if the problem persists.
Another possible reason is that it's just how the guitar string behaves when you pluck it. I hear something that resembles what you describe on my classical (and on other guitars, although less so) when I hit an open string hard, which is obviously what you're doing for tuning. The string naturally wants to resonate at a frequency that has it static at the ends and with a peak in the middle. When you pluck the string though, you're not plucking at the middle of the string, so that pluck has started another frequency with a peak where your fingernail hit the string. Your fingernail doesn't keeping striking the string, so this second frequency vanishes pretty quickly, but it's still there.
Another source of frequency wobbling when you hit a note is that when the string resonates, it's under more tension - instead of being straight, the string is now being forced to wobble 1mm or so to either side. This can pull the note sharper in the same way as a string bend, and it's most noticeable if you pick hard with a plectrum.
McGrath's advice of using an electronic tuner is good. They're dirt cheap these days (£15-20 for a decent one) so no reason not to get one. Make sure you get a chromatic one - the EADGB guitar-only ones are a waste of time.