собираюсь - это всего лишь намерение
about - звучит как уже почти посупил. Is it just me?

So this just means “I’m going to go to law school”?

Yes, this is the only meaning I can imagine - plans for the future, intentions - providing that they get accepted.

Thank you, good to know. For some reason, I always thought it meant that somebody was "about to" do something.

In some context it does.
Я собираюсь уходить ~ I am about to leave
Я собираюсь в дороrу ~ I am packing my things
Я собираюсь навестить родителей ~ I am going to see my parents

In this case "поступить" means to make an application and, possibly, to sit the exams. The application can be successful or not. IMO it was better to say "поступать"

Good to know! Thank you 🙏