Take another look at the original sentence:

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 to fix it the correct way:

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.

Go to the next sentence.

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