Use them? You just play them, right?
When you retune to D Standard, how do you mean "goes out of tune"? Like, you'll tune to D, and then after a few minutes it'll be out of tune? If so, that's just the neck slowly adjusting to the new tension. For a Les Paul style guitar, going from Drop D to D Standard shouldn't be too bad of a change, so I'd suggest going to D Standard, and just keep retuning. It might take it a few hours to fully settle.
At that point, you may want to set it back up if the action is too low (frets buzzing out and such). It's not a difficult process... I'd venture that most of us have learned how to do it specifically because we were all too poor to pay someone else to do it when we first needed it done
. There are a lot of videos on how to do this, something like this...
You don't necessarily need heavier strings... I'm using a set of 10's with a guitar tuned down to Drop B
. They are loose, but I've learned how to work with it. For a more standard tension, 10's would be fine... 11's might be tight, but it's an easy thing to change