Using engagement traffic with google ads?

Im thinking of trying an engagement ad along a google shopping camp, the last 12 months' sales campaigns have dramatically dropped roas is now around 2. I have tried new creatives, proven creatives, warm groups everything you can think off. Now im thinking to stop pushing sales ads and try other methods.

Do sales ads only show conversions if the conversion is made through facebook? As personally i would open the browser search the website and convert directly. Which makes me worry about tracking.

Whats your opinion on this approach?