Just like ultrasawblade said, "since" is not an adverb. In this case, it means "because". The first sentence means
Because I ran, the dog followed me.
The second doesn't make any sense with the comma where it is. If you remove the comma, it means
Because the dog followed me, I ran.
As for your real question about where you can place an adverb, before or after the verb is fine. You could even place it before or after the entire clause.
Hastily, he ran from the dog. (Before the clause)
He hastily ran from the dog. (In the clause, before the verb)
He ran hastily from the dog. (In the clause, after the verb)
He ran from the dog hastily. (After the clause)
I can't think of any examples where the meaning is different when you move the adverb.