Here is the lexicon entry for shin-lamed-mem (the root for "shalom," Strong;s #H7965). If you expand the discussion from the Genesius Lexicon you'll find that it includes discussion of the Arabic root as well. The root implies security, sufficiency, and wholeness and actually does not focus much upon submission. It is summed up well in the blessings of Lev. 26.
To the best of my admittedly limited knowledge and memory of a language I studied many years ago the Arabic roots for Islam and Salaam are entirely different. Like Hebrew the words in Arabic derive from a small set of root consonants with the meanings of the derived words depending on diacritic - roughly equivalent to unwritten except in children's books and in language instruction materials - pronunciation guides as well as various pre and suffixes.
Salaam, a noun, is based on similar consonant roots as Shalom. Both Arabic and Hebrew are Semitic languages (along with Aramaic, the common spoken language of Judea during the time of Christ and still spoken in some isolated, mostly Christian, Middle East communities) with many grammatical similarities. The verb origin for the word "islam" is pronounced "Aslama" (aliph, seen, laam, meem), i.e. to surrender or submit. "Islam" is the imperative form. I do not know what the Hebrew equivalent root or word is for "submit" but it would be interesting to find out.