Let me weigh in here with a few thoughts.
First, with a school horn, if it is at all possible, get the use of the horn restricted to you and you alone. I know that it's hard with large instruments (which are usually only had in limited numbers), but if you can keep other people away from the thing, you have a decent chance to keep it undamaged and in regulation.
School baritones are among the worst instruments of all to keep leak tight and functional. (Although tubas are larger, there are fewer ways to damage the mechanism.) And, if the horn leaves your custody for even an hour, someone without a vested interest in the thing will bash up the low C cut or the long keys on the bell.
(Bass clarinets are even worse. One encounter with a chair edge can nudge a saucer pad on the lower joint, and there goes both your ability to play low notes and to play over the break. Bummer.)
Mind you, you can play a baritone without functional lower keys, but the technique is somewhat "limiting", if you know what I mean.
In the golden days of my youth, when I was foolish enough to take on private students (almost all on bass clarinet - I had a "reputation" throughout southern IL, and they came to me from far afield), I would first ensure that the student was serious about playing. If that was so, then next I would negotiate with the school for an exclusive instrument for the young lady (most of my students were of the feminine variety - weak girl clarinet players get assigned to bass clarinet all of the time). If that was not possible (and it was only possible in about half of the districts), I then strongly urged the parents to purchase a Vito or Bundy horn for their child.
Once this was all in place, the players literally "bloomed" on the instrument. No more squeaking or squealing - little girl fingers which could not seal the lower chimneys on the soprano clarinet were suddenly transformed into powerhouses on the bass, where pads did all of the hard work. The horns "got out of the way", and students were no longer frustrated with their horns.
By the end of the first year, students who were positive about the playing experience were playing out of the intermediate or advanced clarinet (not bass clarinet) method books, ranging well above high C with perfect facility.
I am convinced that this is the only way to go on a harmony instrument. It might cost some bux up front, but if the student is committed, it works like a charm.
Now, as to breathing. As a low instrument musician, you routinely have to move more air through the horn. Even today, some forty five years (sigh) after I first picked up a sax, I still routinely mark breath points through music. Many is the time when I become distracted and ignore these little guideposts (too many low cut cocktail dresses on the dance floor), I find myself at the end of a critical phrase or hold, where the bass of the group is carried only by me and the bass trombone, with "half the air' that I need.
(Example: the Nelson Riddle arrangement of New York, New York, where the band comes to a screeching halt while the Chairman of the Board imitator gets loose and carefree with the "Theeeeese lit----tle towwwwwwn blues" entry. I generally steal a huge, hockey bench breath one bar before the held note, simply so I am sure that the two of us (bass bone and I) can carry it through.)
Leading into breathing here, I have always taught my students to use both the diaphragm and the "auxiliary muscles of respiration" to up their total "blow time". It's a trick I picked up from my juniors hockey coach, back before the dawn of time.
The diaphragm is the normal method that the human body learns to breathe. You "pull" the diaphragm (a large sheet of muscle - in a cow, it's the flank steak yum!) downwards, which in turn expands the bottom of the "chest cavity", causing a partial vacuum that draws air into your lungs.
The auxiliary muscles are the ones that are carried around your ribs. (Think spare rib meat - double yum!.) They work in a slightly different fashion, by pulling the rib cage open to the sides. This too creates a partial vacuum within the chest cavity, resulting in more air being brought into the lungs.
While we reflexively breathe with the diaphragm, even when knocked unconscious, you have to work to get the auxiliary muscles into the mix. The way to do this is to visualize "pulling" the rib cage open with the muscles on the sides. You may have to do this with the diaphragm fully pulled down at the same time - some can only do it that way at first.
Take the deepest breath that you can with your gut, and then - while holding that deep breath, breathe more with the side muscles. You can practice this at any time when you time to spare - my hockey kids were told to do it on the school bus, so as to keep them quiet, but any spare time will do.
Once you master getting them into play, the next step is to practice breathing with them alone. It's hard to do at first (and it's not something you do when you are performing any physical activity), but with application, you will get to the point where you are able to breathe without moving the diaphragm.
Once you get to that point, you are ready to go. In hockey, you play for one or two minute shifts on the ice, followed by twice that amount of time sitting on the bench. After getting to the bench, deep breaths (combining both methods of breathing) are performed for thirty seconds or until the heart rate declines to normal levels, followed then by normal breathing. Works like a charm.
In music, it's a bit different. I generally only use the technique when playing extended exposed passages on the bass clarinet, baritone or bassoon, and it is seamless enough when sitting down so as to go unnoticed (except for the sweeping bass clarinet phrases that I can manage). It gives me enough air to "fill up" the big horns, and it allows for better intonation and playing of piano phrases, all without "circular breathing".
(You will also scare the hell out of the respiratory technician who gives you a spirometer test, the one where you blow in a tube to measure your lung capacity. Up to the point of my work retirement at age sixty, I regularly "blew" 110 - 115% of the predicted capacity, where norms were in the 80% to 90% range. I told my mother that playing bass instruments and playing hockey all those years was good for something.)
The last hint that I can offer is to always spend part of each practice session practicing one level above what you are playing. As with the bass clarinet, baritone saxophone technique is not that different from the soprano/alto horns. If the horn is in good repair (and God alone knows what all is wrong with your Conn), you should not just practice method materials or ensemble parts for that instrument. Instead, "push" yourself to play alto or soprano lines as a regular part of your practice routine.
can be fingered the same as 
: all you have to do is press the octave key. (Yes, I know that that fingering has some tone and intonation issues. Bear with me.) If you're using this fingering OR if you're going down from, say, D to C and don't have any problems, chances are extremely good that there's no problem with the horn, it's with your technique.
is a totally different fingering than 
. Many folks, including me -- when I was learning to play -- have a psychological "break" when jumping these fingerings and (unconsciously) take a gulp of air right before the "break" and then release that gulp of air on the "break" note, thus making that "break" note out of tune or muffled sounding.
