Going from half time feel to double time feel and back

Hi,

I couldn't find anything about this. I'm struggling to make my playback sound ok when going from half time to double time in my score. I used the "quarter note = half note" notation to go from half time to double time (when going on the chorus). I noticed that the tempo (which is set to 62 bpm for my piece) then goes to 120 bpm (which is not the precise double) at the sign. And when encountering the "half note = quarter note" sign to go back to half time feel, the playback tempo stays at 120 bpm.
Maybe I'm doing half/double time notation wrong...
Thanks for your help!