Sentence 1. is ungrammatical. The verb 'piss' needs to agree in number with its subject: 'that pisses'. Sentence 2. is grammatical.1. They always drive over the dotted lines and that piss me off.
2. They always drive over the dotted lines and that pisses me off.
that (subject; relative pronoun, singular)
pisses (verb, singular)
'that' refers back to one clause: 'They always drive over the dotted-line'. You could replace the relative pronoun with another singular pronoun "it", like this,
It pisses me off.
You could also replace the relative pronoun with "The fact that", like this,
The fact that they always drive over the dotted-line pisses me off.
=> The underlined portion functions as the subject. It's one clause, so it agrees in singular number with the verb:
They always drive over the dotted-line = It/That
You could also omit 'The fact', like this,
That they always drive over the dotted-line pisses me off.
Note, 'dotted-line' does not require an -s to express all the dots in the line. It means the line is dotted (i.e., one line that's dotted, a dotted-line).
All the best, :D