It is normal to give an informal offer and wait for you to accept before sending the official offer documents in most recruitment. If you already accept it, it is considered very rude to go back now and ask for more. It is within your right to do so, but be careful of the repercussion.

As for "manager-level" I'm not sure what it means, but anyone who has a basic degree w/o experience joins them as officer, so it is equal to private sector of fresh grad executive. If that is what they mean by "manager-level", then I think is highly misleading.

One thing is NUS admin office has highly inflated job titles. Even their Director pay and responsibility is more like a manager in the outside world. I cannot imagine how an officer can be considered manager.