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!