It doesn't sound dreadful!
If you're listening to yourself and wanting to improve your tone - it WILL naturally improve as you develop as a player. Our musical and technical abilities improve in conjunction with each other, so improving your scales will improve your sound, improving your ears will improve you sound, articulation will improve it, etc. We often think we can isolate areas but playing the saxophone really one whole.
Anyway, philosophy aside, it's hard to tell without seeing you play but I too suspected maybe a too soft reed (perhaps). It sounds like you're getting some good volume which is great, it doesn't sound stifled.
There are exercises you could try to work on control which might help...I'm sure you're bored of hearing people recommend long tones? But they're brilliant. Practice playing from ppp to fff. Practice playing scales very slowly, taking care of each note. Experimenting with overtones.
Sound development is very subtle, and involves minute changes in tongue position, the throat, the embouchure etc. Often simply HEARING a great saxophone player (i.e. teacher) play in front of you can instantly improve you sound! Bizarre but true I think.
Finally, I don't want to mention gear too much, but you're looking for softer, darker, rounder sound you could try a mouthpiece otto link, or meyer type piece with a low baffle. I'm not too familiar with the PTT, obviously they're very high quality mouthpieces but with the baffle I'd expect it to offer projection, guts, brightness etc. which can take some experience to tame.
Good luck! Let me know if you want me to expand on the exercises....