It can probably be a little tricky. What I'd suggest is ... install LW on the new computer, then close LW. Then replace the incomplete folder or its contents with the incomplete folder from the old computer. Then open LW again & it should be fine.

If you attempt to move the incomplete files whilst LW is open then yes you will definitely run into problems! Besides, you still probably have a backup of the downloads.dat & .bak files so replace the newer ones. LW 4.12.4 has the ability to continue old downloads if you resume them from the Library window incomplete folder by selecting the incomplete files & pressing Resume.