The groups you are considering are a special class of Shephard groups, which are obtained from Artin groups by adding relators $\sigma_i^{k_i}=1$, $0\le k_i<\infty$ for every Artin generator. Biautomaticity of some of all these groups is conjectured in http://arxiv.org/abs/0901.0094 and this conjecture is verified in some cases. These results apply to "most" but not all groups that you are interested in.
Misha
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