Which words count for a double word score?
You can do what you want here (score the modified word and the entirely new word).
All that matters is whether you play a tile on a premium square. If you do, you'll get the bonus, for that tile or the word(s) containing it, as appropriate.
For example, if someone plays CLOUD, with a double-word score square just after the end of it, and you play SWORD off of that, forming CLOUDS as well, you'll get double score for both CLOUDS and SWORD. You could also just add the S and score for CLOUDS, but hopefully you can do better!
You'll only get the bonus once, on the turn when a tile is played on the special square. Once it's covered, it's used and it might as well not be there. But it applies to everything relevant on the turn when you do play on that square.
From the rules (in the scoring section):

The score for each turn is the sum of the letter values in each word(s) formed or modified on that turn, plus the additional points obtained from placing letters on Premium Squares.
Letter and word premiums count only on the turn in which they are played. On later turns, letters already played on premium squares count at face value.