Admissibility of heuristic in the $A^*$ algorithm
It's exactly as you suspected:

Sometimes, the algorithm will find an optimal solution even with an unadmissible heuristic... only an admissible heuristic guarantees the optimality of the solution returned.

If you use an unadmissible heuristic, there's no guarantee what will happen.  The solution you get back might be the best one; or it might not be.  And, you probably won't have any way of telling whether the solution you got back was optimal or not, so you won't even know for sure when this is a problem and when it isn't.

If you use an unadmissible heuristic, you'll often (but not always) get sub-optimal solutions.