Shoe covers (or other tricks) for commuting in shorts
if I wear the same clothes and just go for a walk/jog, then my feet stay dry for much longer


Do you have mudguards (fenders) on your bike? If not your feet are in the spray from the front wheel and will get wet unless you wear over-boots. Or gumboots.

For cycling when the roads are wet mudguards make a huge different to your comfort. You'll stay dryer because the rain will only hit you from above, rather than spraying up from your wheels as well.

Gary.Ray's answer deals well with the actual shoe cover question. What I would add is that plastic bags and rubber bands work well as a short-term solution for one trip, but they don't last long. But that might help you decide whether shoe covers are worth having at all.