You are obviously correct in saying that the absence of the APU would have a marked effect on handling.
There are lots of situations in which engineering actions change the basic weight and balance of an aircraft. These include the embodiment of some types of modifications and also some types of structural repairs. The removal of an APU would fall into the same category.
Whenever these type of action are carried out the required paperwork includes the recalculation of the BEM, DOM and DOI. So if the engineers are doing their job properly, the loaders and pilots just use an updated load and trim sheet and complete it in the usual manner.
So how would I correct this weight and trim on loadsheet, ?
As I have said above, if the engineers are doing their work properly the loaders and pilot do not update the DOM and DOI in the loadsheet. They simply use the updated figures that have been provided by the engineers.
or verify everything if a corrected loadsheet was handed to me
Again, you should not need to check it. Do you check the calculation of the BEM, DOM and DOI for every flight after any significant engineering work has been done?
I suppose that there might be a situation in which you had reason to suspect that the figures had not been updated. If, for example you flew the same aircraft every day and had become so familiar with the DOM and DOI that you could remember them. Then if the APU was removed one day and you spotted the fact that the figures in the load sheet had not changed, you would obviously need to have some checks done. But to do this you would need to go back to the engineering records, so it's not the kind of thing that you could do on the ramp.