My recently acquired (and on close inspection, genuine) TinySA with 1.1-74 firmware exhibits this behaviour. I discovered this by chance with my VX-7R, having found the carrier would disappear when I selected FM modulation, but not with AM. I happened to use the channel search function and found where the carrier is going.?
On low output with FM narrow or FM wide modulation, the carrier shifts up 160kHz while the displayed frequency remains as selected. With AM the carrier remains correctly on the selected frequency:

Low output, no modulation

Low output, 1kHz AM

Low output, 1kHz NFM

Low output, 1kHz WFM
On high output with FM narrow or FM wide modulation, the carrier shifts up 80kHz while the displayed frequency remains as selected:

High output, no modulation

High output, 1kHz NFM

High output, 1kHz WFM
The TinySA was delivered with a few days ago with firmware 1.1-12 (or thereabouts) and if my memory serves, it did not then exhibit this behaviour. Maybe someone can do a sanity check and see if this is indeed a bug.
Regards,
Liam