"자고도" is not a fixed form but a construction of -자고 + 도.
-자고 is a quoting suffix for a suggestion or invitation, combining -자 ending with quotation suffix -라고.
- 이제 가자 = Let's go now.
- 영철이가 "이제 가자"라고 한다 = 영철 says "Let's go now". (direct quote)
- 영철이가 이제 가자고 한다 = 영철's saying that we should go now. (indirect quote. "가자" + 라고 -> 가자고)
If your example sentence didn't have 도 in it, it would be:
- 계속 사귀자고 안 할 거야? = (너는 나한테) "계속 사귀자"라고 안 할 거야? = Aren't you going to ask me to keep seeing you? (straightforward question about the person's intention)
* 할 거야 = 하다(do, say) + -ㄹ 거야(intention (I will, I am going to) = will do/say, am going to do/say ...
* 할 거야? = will you? are you going to (do/say ...).
* 안 할 거야? = won't you? aren't you going to (do/say ...)?
도 adds the nuance of: even if, in addition, as well, or either.
It treats the thing or idea as so appropriate that it's a matter of course, giving it emphasis.
So the sentence means "Aren't you going to ask me to keep seeing you either?" (like "on top of all the other things?"), or "Aren't you even going to ask me to keep seeing you?" (like "I might understand other things, but this?"), which are in the end more or less the same.