Boots! You know you want them. You want them for the reason Brakes gave above. Overshoes wear out. I was just about to by my 4th set when I thought spend yer money wisely on something that will last.
Northwave Celsius. Warm, dry, comfy. I shudder to think of going back to summer shoes with overshoes. They leak and your feet get cold. Then you feel miserable and your ride is ruined. NWCs are going strong after a couple of years. I do look after them well (as I do with walking / climbing / winter mountaineering boots.) I hope to get plenty more years out of all my boots.
Buy boots! Try them with mini-gaiters when it’s really gash. In the past I’ve had leakage into the boots down the legs of tights. I have Endura trousers over gaiters then gaiters over boots. Bone dry in pretty much everything Winter has thrown at me.