I did a little more digging around and I ended up haphazardly stumbling on the answer. I was missing "Integrated Security=SSPI" in my connection string and it turns out I didn't need the dot before "\SQLEXPRESS" in my data source. Here's the connection string that worked for me:
adodbapi.connect(r'Provider=SQLOLEDB;Data Source=COMPUTERNAME\SQLEXPRESS;Initial Catalog=Test;User ID=COMPUTERNAME\USERNAME; Password=PASSWORD;Integrated Security=SSPI')