You have discovered one reason why business logic should usually go to a business rather than to access data at a level. Of course, there are exceptions due to performance and sometimes security considerations, but they should remain exceptions.
Having said that, you can still develop a testing strategy for your sprocs (although depending on how extensive they are, it may or may not be right to call these tests "unit tests").
You can use the unit testing platform anyway.
In the initialization section, restore the test copy of the database to a known state, for example. downloading it from a previously saved copy.
Then run the unit tests that execute the stored procedures. Because stored procedures usually do not return anything, your unit test code will need to select values ββfrom the database to check if the expected changes were made or not.
Depending on the possible interactions between stored procedures, you may need to restore the database between each test or between groups of related tests.
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