I used to play the trombone religiously when I was in school but during my second year of college I had to give it up. I haven't played in a group or on my own for more than a couple hours over the last 5 years. I've recently decided that I'm going to become a music teacher so I've started practicing again but I feel like I'm not progressing in regaining my skills. I've been practicing for almost a month now and my tone is still rough and my range is still limited. Any tips to help improve while not damaging my chops?
On string instruments, we often play open strings in order to practice bow control. My intuition says that the analog for brass would be to use whatever slide position is most comfortable, and play different partials (start with 1st, play long notes).
Try to keep your tone as even as possible. This isn't an exercise in how long you can hold the note, but how consistent you can keep the note while you're playing. Manage the air flow, but don't constrict it.
There's two main angles here, what you're practising and how you're practising it. For the what the rudiments and fundamentals are always key be you a new beginner or a seasoned pro. Many folks who come back to their instrument after a long break feel like they should be able to play like they used to, something your dexterity and control might agree with. The exercises that you first used to help build good tone, endurance and flexibility, and musicality before you took your break will be the best things to get you back into shape. For trombone this probably means developing endurance across the full range of the horn, refining your intonation, and balancing legato and powerful playing.
As for the how, it's usually best to spread out your practice across as many days as you can manage. If you only have say 3.5 hours to practice a week you're better served practising 30 minutes a day rather than 3.5 hours in one go. This is even more the case for brass instruments as your embouchure can naturally deteriorate quite quickly. It's also better to get less high quality practice than lots of distracted practice. Chances are in your adult life you have less energy and focus to dedicate to music than you did when you were younger leading to less productive practice sessions. Anything you can do to bring focus and positivity to your practice time will help.
Finally, I know it's corny, but try to enjoy it! People learn more when they're having fun and you're more likely to want to dedicate time to your playing if it's a pleasure. Be it playing with others, noodling along to your favourite recordings or whatever, if you're looking forward to getting that horn on your face you're going to learn more. Good luck with the audition, you've got this!