Here is the item again:
To get on his mother's good side, (A)
Rufus offered to unpack the groceries,
(B) but dropped the carton
of eggs that Mom would later need (C)
to make chocolate-broccoli muffins for the church bake sale.
You correctly realized that the comma after groceries was wrong. But dropped the carton of eggs is a lonely verb phrase. Such a phrase does not require a comma when you attach it to the main clause in front.
You chose to do this:
To get on his mother's good side, Rufus offered to unpack the
groceries. But he dropped the carton of
eggs that Mom would later need to make chocolate-broccoli muffins for the
church bake sale.
Providing the subject he for the verb dropped and then making this subject-verb pair a separate sentence is an excellent way to fix the problem. Well done!