First, I'm aware of this question: "Let's" vs. "lets": which is correct?
What I want to know is if there are other similar constructions formed by contracting a verb with "us".
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No there are no other contractions of words with us in the above specified form as in other contexts contractions ending with 's refer to contraction for is, has or does which is very informal and hence rarely seen. Let us is a special case that is used in forming the imperative mood in first person singular for Let us.
Here is the link to the reference in case you wish to find further clarifications.
Hope this helps. :)