Review the original item:
Richard squeezed, thumped, and shook the empty shampoo bottle, hoping for
a few drops to wash his dirty hair,
since the bottle was completely empty, Richard worked up a lather
with a few sprinkles of bathroom cleanser instead.
You chose the correct way to fix the error:
Richard squeezed, thumped, and shook the empty shampoo bottle, hoping for
a few drops to wash his dirty hair;
since the bottle was completely empty, Richard worked up a lather
with a few sprinkles of bathroom cleanser instead.
Richard squeezed, thumped, and shook the empty shampoo bottle ... begins a complete sentence. Since the bottle was completely empty, Richard worked up a lather ... starts the second complete sentence.
You are always correct when you use a semicolon between the two complete sentences that create the comma splice.