How to screen the game for possible draws?
There are only a few such possibilities:
- Perpetual check
- Stalemate
Repetition. When the winning side cannot find or choose the right strategy to attack - there is a good chance to force him to repetition.
a third option is to try to force a materially-drawn position, like a drawn endgame (like rook vs minor piece), or an endgame with a fortress.
I'd first check for stalemates -- that just requires a quick scan of the board. If your pawns are locked up, and you can sac your pieces, and your king has no moves...you can usually see very fast if that is an option.
next I'd look for perpetual, but you usually find those by just calculating your normal lines. you should be able to tell pretty fast if you can force perpetual (if you have no checks you have no perpetual!).
forcing a materially-drawn position is much harder as it requires knowledge of endgames and fortresses. but you can go a long way with some basic endgame knowledge like the "wrong color bishop", and other basic piece relationships. Here's a good list of some fortresses. Learn these, and you have another drawing resource in your toolkit: https://en.wikipedia.org/wiki/Fortress_(chess)