coolant leak below the oil (filter) cooler

my 99 lumina blew up a coolant leak in the area connecting the hose and the oil cooler. i fixed it up by trimming the hose and clipping it back on securely. now, this part has stopped leaking but i notice a new leak underneath the oil cooler/oil filter assembly. could it be that the oil cooler itself is shot with rust? i also noticed that after this incident my exhaust has started blowing white smoke when i start up in the morning, when there was nothing of this kind before. does this mean that oil has entered the cooling system or water has contaminated the engine oil? can a worn out oil cooler really do this damage?

any insight and info would be much appreciated. thanks lots.