Is a separate execution plan generated for each dynamic SQL query that is executed within a stored procedure? If I have a stored procedure and I re-write the logic so that some of the queries are executed as dynamic SQL, will those queries now get their own individual execution plans in addition to the stored procedure getting it's own execution plan?

If so, could this be potentially more globally efficient on my server if there are other ad-hoc queries that exactly match the dynamic SQL queries in my procedure (or even if there are other procedures that use dynamic SQL to call the same exact queries as well)?
Yes, the dynamic SQL you submit will be treated as ad-hoc. I.e., a hash is generated from the text, making it sensitive to everything (space, upper/lower, search arguments, etc). 

If you happen to submit the exact same string as a regular ad-hoc (not dynamic sql from a proc), or from other procedures as dynamic SQL, then the plans can be re-used.

What is best for you, we can't say. An option for you is to use sp_executesql and parameterize your queries to allow for plan re-use. OTOH, you will now have parameter sniffing. Either "pay" for plan generation for every execution or re-use plans. Your choice.