"I am tired and cross."
Translation:Tha mi sgìth agus crosta.
Yes, and so does tha mi.
is mise is used when sentences like is mise Iain "I am Iain", is mise an t-ollamh ùr "I am the new professor", is mise am fear as òige "I am the youngest one". What these have in common is that "I" is described using another noun, which is the domain of the copular verb is and its various forms.
tha mi as in this sentence tha mi sgìth agus crosta "I am tired and cross", and also in tha mi ag obair "I am working", tha mi an-seo "I am here" links the subject "I" with something that is NOT a noun.