Is this usage of "woo" proper?

Solution 1:

Alternatives would be "kowtow to" or "curry favour with", "fawn over", "brownnose". And my parser broke on "acting a bit overly".

Solution 2:

I think you're right. 'Suck up to' might convey the intended meaning rather more clearly, depending on the degree of formality of the context. I find the use of 'overly' in this context unusual. In British English we might substitute 'over the top'.

Solution 3:

In this case, I would say that woo is a good choice, because of the objection you raised. Because of the primary meaning, it carries that extra nuance which insinuates that the other students are acting too affectionate towards the teachers.

And I agree with @Barrie that "overly" is out of place here, but I would say it is dead wrong, unless it is an acting class. The word "acting" is a linking verb here, so an adverb is inappropriate. AmE would also use "over the top", too, by the way.