Traditionally, if you moved the entire incomplete folder including the downloads.dat & downloads.bak files across to the new computer then it would work. However the new version LW may not use those files to store information any more not sure. BTW that process could only happen whilst LW was closed. So idea was, you installed LW on new computer, closed LW. Then replaced the new incomplete folder with the old one (else the files inside as long as it included the downloads.dat & downloads.bak files.) But emphasis was LW Must be CLOSED when this happens else it will wipe the old incomplete file history data.
I'm still using LW 4.16 & it still uses the downloads.dat & .bak files. However I think LW 4.18 no longer uses them. I have not found out where the data is stored for LW 4.18 because I cannot run that version on this particular computer.
OH ... and after opening LW after exchanging the incomplete folders, if the old downloads do not show in the download window then you 'might' need to go to the Library window & select the incomplete folder, then select all the incomplete files & press Resume button which will force them into the download window. You might be best to do a % at a time if you have too many. Using that technique if you need to use it, the files will probably say zero % until they actually start to download again upon which time their previous download history will show.