In my limited experience with them, Tecumseh engines are mechanically robust and durable, but tend to have terrible carburetors, which spoil the whole picture and make you want to kick them rather than fix them.
Because your last batch of oil had sat overnight in the engine before you drained it, you should not feel too confident that you got all the water out. Soluble oil forms a durable emulsion with water, but engine oil does not, so what looks like a chocolate milkshake right after you've run it, might turn back into oil with a pool of water underneath it, by morning.
I suggest you finish off the job on that first mower by running it for 10 minutes or so, then draining the oil immediately into a clean container. If it is clean not milky, you can put it back into the engine and feel fairly confident. If it looks milky, you still need to get that soluble oil. You need to get it anyway, to deal with the second mower without using up many litres of engine oil along the way.
