Did I misread your compatibility mode? If so, my bad, apologies. Yes, you can have the new compatibility mode enabled and then disable the new cardinality estimator.
As to the rest, you say hundreds, then you say thousands. Each is going to behave a little differently, so which is it? It matters. Two hundred rows and 3 changes in a week, stats should be identical. Several thousand rows and 3 changes in a week, stats should still be identical. However, thousands of rows and more changes, then auto stats will fire. Row estimates will change and using the legacy cardinality estimator, they should be roughly the same.
The question is, where and why are the estimates off. Without seeing the plans, I can't even guess. The optimizer is different from 2012 to 2016. It's not just the cardinality estimator that's different. This could lead to differences in behavior. Also, the mechanisms used to gather statistics are a little different in 2016. That also might lead to some behavioral changes. However, without seeing the details of the setup, all I could do is speculate wildly. If you at least post a before & after plan, one of the ones that especially egregious, so it's possible to evaluate what's going wrong and why, I could add more than vague suggestions. I mean, are you seeing the histogram used to arrive at row counts, or is it just using the selectivity of the statistics to arrive at them? This can change behaviors. It's just too hard to know based on too little data.