What does it mean in Ukrainian/what is its origin?
big (and and fat) - of people - in Polish it has figurative or extended meanings don't know about Ukrainian
It is a polonified form of an originally Ukrainian surname
it's not even Polonified on the outer the sounding of the surname would be perfectly the same in Ukrainian
the form Tugi for a male and Tuga for a female mean that the originally the surname was an adjective - it hardly fits your version of it being somehow connected with worry