Q. (See question above) Can one use cologne or liquid perfumed deodorant applied directly on the body during Shabbat, if it also passes on by contact an aroma to the clothing one is wearing?
A. Mishna Berura (511: 27) writes that the prohibition of molid or creating fragrance on clothing and utensils is only Rabbinical. Therefore, if there is no intention on perfuming the clothing, rather just removing bodily odors, Shulchan Aruch Horav (511) permits using it.
Shemiras Shabbos K. (13; n. 82) rules that one may use a handkerchief or other garment that has been perfumed before Shabbos.
Horav Shlomo Miller’s Shlit’a opinion is that it is permitted.
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a