In the English Vocabulary Profile (EVP), ‘Still‘ is listed at B1 with the meaning ‘despite that‘, where ‘that’ refers back to the previous thing talked about. This is slightly different to:
‘despite‘ as a preposition, which is also listed at B1 with the meaning:
without taking any notice of or being influenced by; not prevented by
All the EVP examples follow either contrast markers: ‘but‘ or ‘even though‘.
The following example is B2 because of the complex conjunction and adverb with the modal verb.
Cutting Edge Advanced has ‘Still,‘ at the start of the sentence as a discourse marker. This can indicate contrast to what has gone before. ‘Still’ in this position is also considered to be a ‘conjunctive adverb’. For example: