Before revision:  Only very recently \mbox{%DIFAUXCMD DBLP:journals/corr/abs-1907-11422,DBLP:journals/corr/abs-2002-07152 extended the classical +2 (respectively, +4) spanners for unweighted graphs of size O(n^{3/2}) (resp., O(n^{7/5})) to the weighted setting, where the additive error is +2W (resp., +4W).
After revision:  Only very recently ABSKS20 extended the classical +2 (respectively, +4) spanners for unweighted graphs of size O(n^{3/2}) (resp., O(n^{7/5})) to the weighted setting, where the additive error is +2W (resp., +4W).