Jen, what you have ahold of there is a wobble. They can sometimes be hard to pin down, particularly if it were happening on just one note, like low D or B. But since you have problems with multiple notes, it might be a more straightforward fix, yet would likely require a good sax tech.
First thing to check is leaks. Tricky to leak light a bari, but you could get one of those flexible hose-like things with lights inside. All your finer hardware stores have 'em. Get one long enough to shove in the bell and push all the way up to the upper stack. Check all around each pad. Your leak might even be the C key (which is a key without a keytouch, above the B key. Could even be the A (middle finger, LH) key which depends on a piece of felt being thick enough. Point is the leak could be anywhere!
Also important is the fit of the neck tenon into the receiver. Should tighten up snugly.
Also, there should not be gaps between the mouthpieve and cork. Can get oddball problems if the mpc is loose!
How is your tuning with that mouthpiece other than the wobbles? Check with a tuner. I have seen low range wobbles caused by the horn not adjusted to true pitch! Odd, but very true.
In the case of wobbles, it's very common to have "good" notes further down the horn.
More odd wobble trivia from the world of physics -- Wobbles can be caused by "standing waves," essentially sound waves at certain frequencies being reflected by the bow back up the horn, colliding with sound waves being produced by the mouthpiece. It's an acoustic anomaly that I've heard of being solved by placing a cork (or the upper stack end plug) in the bell. I have also heard of techs installing a small baffle in the bow to counteract this phenomenon! Sounds far fetched, but true!
In your case, leaning toward one or more leaks, or loose neck and/or mouthpiece, or maybe even the position of the mpc on the cork. (Just for the heck of it, move the mpc to different positions on the cork and see if the problem goes away. Also try different reeds or mouthpieces.
Having said ALL that, leak light it and let us know.