These are conjunctive adverbs. It is true that "still" is just a normal adverb also, having the meaning "ainda", and sometimes, "até". Generally, conjunctive adverbs join 2 independent clauses (sentences), but sometimes, they occur at the beginning of sentences to join 2 subsequent thoughts. Here are some examples :
1) I don't have any money; nevertheless, I will find a way to travel to Europe this summer.
2) I don't have any money; still, I will find a way to travel to Europe this summer.
3) I don't have any money. Nevertheless, I will find a way to travel to Europe this summer.
4) I don't have any money. Still, I will find a way to travel to Europe this summer.
One thing to note is that "nevertheless" and "still" are used to join 2 contrasting thoughts. Also, pay special attention to the punctuation in each case.