If it's smoking heavily with white smoke, you are on the right track by investigating the coolant. Moisture in the exhaust system will cause white smoke too, but that goes away after too long. It seems the best thing to do for starters is to run a pressure test on the cooling system to see if it leaks down. Also monitor coolant level. If this confirms a loss of coolant, which is making its way to the combustion chamber, then you need to find out where the leak is coming from.
If you knew the problem was due to burning water because of the white smoke, then all the work you did with rings, valves, and seats was for nothing. I'm sorry to hear about that, but it is best to test and evaluate up front.