Trumpet advice

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!

<Added>

Sry, about double post.

Trumpet advice

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