I am wondering if you noticed -겠-(끝마쳐야겠어요) and -(해)야-(끝마쳐야 해요/돼요) in each phrase.^^
"-겠-" indicates that someone is ready/willing to do sth or it's time to do sth.
(*sth = something)
그만 집에 가야 겠어요.
= I'd better get going.
= I should be going.
= It's time to get going.
이젠 끝마쳐야 겠어요
= I should finish it.
= I will finish it.
= It's time to finish it.
Whereas "-야-" is used to mean "need to do sth or have/has to do sth" so the second one will be "I need to finish it" or "I have to finish it"
나는 오늘까지 숙제을 끝마쳐야 해요
I need/have to finish my homework by today.
= The homework needs/has to be done by today.