No, you can't say "shizu ni". If you desire to use "shi" anyway, you can say "shinai de". For example, you promised to call her but you went to bed without calling her. Denwa sezu ni neta. Denwa shinai de neta. Same meaning.

Basically, "zu" is "nai" in classic Japanese. Although it was mostly replaced by nai in modern Japanese, still it remains here and there. Use the stem for nai-form in order to make -zu. So you don't need to edit the conjugation table. (Suru is exception.) Taberu(eat) - Bangohan wo tabenai de neta. =Bangohan wo tabezu ni neta. (I slept without eating dinner.) Yomu(read) Hon wo yomanai de kansou wo kaita. =Hon wo yomazu ni kansou wo kaita. (I wrote some review without reading the book.) Motsu (have something with you) Mizu wo motanai de yama ni nobotta. =Mizu wo motazu ni yama ni nobotta. (I climbed a mountain without having some water with me.)