Although you don't want to remove the ceilings and hang drywall, that is your best bet. You cannot cover the groves because the tiles actually move a great deal. Most are made of a paper or ground wood and cellulose material with binder. Any moisture swells the tiles quite a bit, so spackle does not stick. Paint is also a product that causes them to swell. The tiles themselves are not flat and most display a convex shape in time. This further complicates hiding them with texture and paint.
The tiles are probably covering a ceiling in poor condition. Fixing the root cause of the problem is really the best way to go. You might do one room at a time, but removal of the tiles and the furring strips and ceiling behind them is probably the best long-term solution. If they were in good condition you wouldn't be proposing to cover them.
The difficult part of this project is not the installation of new drywall, but the demolition of the old tiles and plaster ceilings. It may be possible to apply drywall over the damaged plaster after the tiles are removed. I have first-hand experience with your problem and was able to do a similar project. It is a lot of work, but the results will be worthwhile.