Don't say "oftener". This is "more often". So you would say "ever more often". For the first phrase you could say "ever more often", "increasingly often", or "more and more". For the second phrase, "less and less" is the preferred phrase. However, you could say "decreasingly often", or even "increasingly less". Or you could switch the sentence around and say "She FORGETS her native town ever more often". However, I have never heard the phrase "ever less often", so don't use that, even if it would be grammatically correct (and I'm not sure that it is).

To say something happens more often you can use 'more often/frequently' or 'increasingly often' or 'more and more' but these expressions usually go at the end of the sentence/clause. 'He stays at work after the end of the working day more often/more frequently (than he used to) increasingly often'. 'He stays at work after the end of the working day more and more.' To say that something happens less often you can use 'less often/ less frequently' or 'less and less'. She remembers her native town less often/frequently (than she did before) 'She remembers her native town less and less. BUT remember is a stative verb and we don't normally say we remember one particular thing less or more often. So 'less and less' would be better with 'remember'.

He ___ stays at work after the end of the working day - just use increasingly as its meaning includes often.... He increasingly stays at work after the end of the working day, and She ___ remembers her native town. I would change remember to Forget.. as the forgetting is increasing.. not remembering.. She increasingly forgets her home town. My 2 cents for what its worth... cheers