How I see it:
They solve their problems getting drunk - They solve their problems when they get drunk/stoned.
They solve their problems by getting drunk - Getting drunk/stoned is the way they solve their problems.
Drunk always means having had too much alcohol. Stoned is usually used for marijuana. There is some flexibility in the meaning of stoned, you could use it for alcohol if the context made it clear that alcohol was the drug in question, but it might still sound a bit strange. The two words are not interchangeable.