Question learning soprano & set up

Hello everyone!

I'm a 7 year alto player in highschool, and my director is having me learn soprano sax for solos in Lincolnshire Posy. This is my third week working at it, and it's been awfully tough, as it seems the transition is for everyone, but I'm having a few issues that I haven't heard anyone talk about.

I'm having a lot of trouble on some of the low notes. low E, Eb, and D all don't really want to speak well. It feels like I really have build up a lot of pressure and force air through the instrument to make them play at all.

middle G is also extremely sharp, which I found quite odd!

The school is lending me a 1978 Yanagisawa soprano (not sure of the model), as well as an Otto Link Hard rubber 5* mouthpiece and what appears to be a stock ligature with that. I'm also playing on 3 1/2 Hemkes right now.
Obviously the set-up is very...odd...until I buy new equipment, but could that affect any of the issues I'm having?

I'm completely new to this world of things, just thought I'd check if the trouble is all me or if the horn isn't helping things much!

Thanks!