Break the rust with penetrant and hitting the screw with a hammer, through the screwdriver or not. Can tap the area around trying to get penetrant inside. Can twist the screwdriver both forwards and backwards to try to get it to break free. Make sure the slot is cleaned out because if it's full of dirt the screwdriver can't reach full depth and will strip out.

Make sure your flat blade screwdriver takes up at least 75% of the slot of a flat screw. More snug is better Make sure the phillips screwdriver fits neatly and completely into the screw. People make the mistake of thinking that all phillips are the same. They are not, you simply can often force it. You need 4-5 different standard sizes for ideal fit to not strip everything.