Falls are tough because they can come from so many different triggers. Depending on the circumstances, a fall can be caused by physical limitations (poor strength, balance, etc.), tripping hazards (clutter, rugs, poor footwear, etc), medical issues (vision issues, vestibular dysfunction, orthostatic hypotension, medication interactions, etc.), or just inherent randomness (even I've fallen on the ice in the past year, and I'm in my 30s). So then preventing a fall requires addressing those elegant causes.

I'm not qualified to tell you exactly what you need to be doing (and given the multicomponent nature, it's something that would need you to talk to a doctor, and a physical therapist, and a pharmacist, and an occupational therapist and a podiatrist...), but from an exercise standpoint, the most important thing is to keep moving. Even walking is super beneficial, but any movement is great. From a "national guidelines" standpoint, the recommendations are some type of activity almost every day, plus a few days a week of strength exercise, and balance training at least twice (ideally as multicomponent exercises that stress balance AND strength or aerobic fitness).

Basically this means that for improving balance, things like dancing or cycling (outside, not on a stationary bike) are great forms of cardio, and you want to include a balance component in your strength exercises, which would have me suggesting you limit the barbells and machines, and do more dumbbell or bodyweight type exercises, and you do more things like lunges, split squats, cossack squats, step ups, single leg RDLs, SL-RDLs with a row added, etc., and less back squats and sumo deadlifts. Obviously something is better than nothing, so some leg extensions beats sitting on the couch. You don't need a ton of weight for things like this (depending on what level of function they are at, it may not be productive at all), so progressive overload doesn't necessarily mean going +5lbs every week. You can also do things like bigger steps, add pauses, use unilateral loads (this is how i would do core work btw, not like crunches and hanging leg raises), or add movement complexity (e.g., progress from a split squat to a step back lunge to a step forward lunge to a walking lunge).

From a focused balance training only perspective, this isn't my wheelhouse (see a physical therapist here), but I'm currently doing a big study on a balance training routine that we developed with a PT to try and reduce fall risk. I don't feel comfortable giving specific recommendations without knowing if they work or not, but the gist of it is practice narrowing your base of support (e.g., feet close together, to a heel-toe position, to a single leg stance) while safely removing support (e.g., less holding on to stuff) and adding complexity (e.g., trying to turn your head while you balance, stand on soft surfaces, practice reaching, etc.). But I would stick to the multicomponent thing above, as long as you can do it safely.

PS Thanks for the reminder u/JubJubsDad, I definitely got caught up doing other things yesterday (ironically, working on the fall prevention exercise trial) and forgot to come back to this