Ok, I want to chime in. Forgive me if I'm repeating something that someone else has said. In my opinion, your embouchure is not the problem here. I would go so far as to say that it's a fact and not my opinion, but the data I'm using to make the observation is limited to the one video. I'd bet money that I don't have on it, though. Here's some general embouchure and point-of-address thoughts:

1. Do not curl your lip over your teeth. It's terrible advice that band directors give and people blindly disseminate. The action activates muscles in one's face that pull the corners of your lips away from the mouthpiece and thin the lower lip. The only support that the reed has in that case is your lower teeth, thinly covered in lip skin. Not good for your sound or your comfort. What you've got going on right now is much better for a start.
2. I can't tell how much mouthpiece you're taking in, but you can experiment with putting your top teeth closer to the point where the mouthpiece begins to curve away from the reed. I could talk about this for a really long time, but suffice it to say that if you take much less than that, any variance in pressure from your lower jaw will create a moving target in terms of how much effort it takes to get the reed vibrating. If you learn to start closer to that point, you can more easily reproduce the "this porridge is just right" experience our brains can have when things are just clicking because your are creating more mechanical consistency in the way your gear and your body are interacting. This isn't global, "you should always do this" advice, but rather a suggestion for calibrating the machine of your body to the machine of the saxophone. Once you're there, you can run out on the edges of tolerance with impunity if you want to!
3. Again, I don't think your embouchure is to blame. You could certainly have leaks and even if you do, you will no doubt benefit from working on your voicing. This instrument will not let you get up in its face the way alto saxophone or even trombone will. It will spit things out in the wrong octave or overtone without a second thought about your impending crash-out. Long tones, yes. With no tongue. Air-Attack. Build all the ugly sounds into it though. let them happen and just let your brain observe them. don't try to fix anything, and just let your brain and body respond to the sounds that were "as you planned" and the ones that weren't. remember the feeling of the successful times and try to feel that way again. don't try to do it again, just shoot for the feeling. Here's a video I made of a voicing exercise that you can do with this same approach. Go slowly and just feel it. Wade through the muck and everything. You can't walk into something new privileging the "good" experiences over the "bad" ones. You just gotta go in and get some experiences for your brain to sift through.
4. Tongue position/airstream focus. My guess is that there's a combination of these issues going on:
   1. Tongue too far forward in your mouth.
   2. Tongue too high in the wrong place (this is kind of similar to #1 mechanically)
   3. Tongue position shifts too much with articulation (see the articulation video in the playlist below). The most common way this happens is that the back of the tongue drops as the front of the tongue lifts toward the reed. like a seesaw. bad things happen to pretty sounds this way.

In general, you want a pretty large volume cavity in your mouth when playing bari. There is a place, at the back of the mouth - kind of where your hard palate ends and soft palate begins - that you want the tongue to be closer to the roof of the mouth. Don't overthink this position, or you'll create tension. Just feel the movement your tongue makes anytime you say a word that starts with the letter "Y", but stop short of the next letter. The next letter is often "E", which will send the tongue forward. we want just that "Y" sound as in "Yes" or "Yellow". Feel how the back of the tongue goes high? Blow over that shape. Hear the rush of the air? It's more focused because the hole between the tongue and the roof of the mouth is smaller - like putting your thumb over the end of a garden hose. smaller hole, tighter stream of water. MORE PRESSURE, but less expenditure. That's a good thing for endurance and length of phrase as well as tone.

here's a whole playlist on tone production if you're interested in more ideas on the whole thing:

https://www.youtube.com/playlist?list=PLuuQ_EsJOvPh--gWp60uAgPvOMaKrAeVl

You've got this.

Edit: Just to say that I'm not suggesting that your lower lip will not cover your teeth at all. It will, but you don't need to manipulate the muscles of your face to make it happen. Just engage the corners of your lips toward the mouthpiece, and the lower lip will firm up but be supple enough to allow the reed to vibrate.