Do I only discard down to 10 after drawing 2 cards on my turn?
Yes.
Your hand limit only applies after your draw step.
From the most recent rulebook, &quot;This Is How You Roll&quot; turn order on page 5:

4. Draw 2 Cards
Just what it says: draw 2 cards. If you need to draw, reveal, search for or look at a card and your deck is empty, shuffle your discard pile. Put it on the table face down — that’s your new deck. Start drawing from there.
The maximum number of cards you can have in your hand at this time is 10. If you have more than 10 after drawing, discard down to 10.