The following sequence is how Stash is working in my spreadsheet. I have 3 cards left in hand. I stash 1 by moving it from hand to Stash, then click discard/draw, discarding 2, draw 4, and put the stashed card back in hand for a total of 5 cards in hand.
So based on the clarification in the rules thread, this is working as intended.
So far the only idea I have for handling Shrine better is to create a checkbox for "Shrine Built? yes/no" or a Shrine Draw button. If I change the draw 1 button, I'll probably run into a conflict with how tech choices are saved.