I've been playing alto-sax for 4 years now and recently I've decided to become a multi-instrumentalist and play trumpet as well. Well, it hasn't been working out for me to say the very least.

Here are the positives:

I can make a sound

Thats about it for the positives.

Here are the negatives:

My sound is very "raspy" and "airy".

I can't buzz without the mouthpiece or at least produce a good buzz.

Whenever I play, my mouth always opens and I produce a raspberry sound with spit flying out of my mouth.

I'm also wondering whether the mouthpiece has anything to do with the sound I'm making. Should I buy another one?
The trumpet that I'm trying to play is a new Yamaha model, btw.

So if you have any advice for me, I would really appreciate it a lot, TY!

get a teacher if you can

A change of equipment wont help anytihng, unless yours is an extreme case, such as starting as a beginner with an asymetric mouthpiece, or something similar.
The best thing is to get a teacher.

the problem is when I buzz, the buzz comes off-center to the right and is not directly in the center of my top lip. I don't know why this happens...