Get rid of the nylon, it chafes plus it’s too wide for her to feel any nose pressure. Switch to leather, preferably with the padded pencil bosal. Be absolutely sure that bridle is adjusted properly so there’s no excess leather flopping around.
Something is creating that head shaking, biggest culprits are loose leather and sometimes noisey snaps and buckles flopping around or something is chafing and I had a horse that the brow band pinched the bottom of his ears. Check that too.
What bit do you have on her? If it’s moving aroundkr has loose shanks or parts. it could be triggering the head shaking. It also might be too wide or too narrow.
These things usually don’t bother them slow but when they move out, there’s more body and head motion that will make anything in her head move more.
Put on your detective hat and THINK about what’s making her shake her head only at the run. Something is going on. Your job to figure it out. That’s the secret if training, try to see and feel it as the horse does.
The tie down won’t help the head shaking, they can shake their head fin in one, but if OP wants to continue using it, got no problem with it as long as it’s a proper fit, doesnt rub ( as I think the nylon might be doing) and it’s not just thrown on to crank the head down, which just gets them bracing on it and block proper use of the shoulders in tight turns so backfires as far as BR goes.
This is a 21 year old game horse veteran too, who may always have worn one but seems to have only just started head shaking so that’s where OP needs to look and think, something is bothering her that was not before. Be happy to continue to help her learn how to think this out and solve this.
Never regularly used a tie down on my Western horses except one, that I didn’t own, who came as a head flipper, needed one to keep his head out of my face, literally, until that got schooled out, the others didn’t need one, but they were cheap projects and a few needed a little, temporary help with bad habits.
After I switched to Hunt Seat, I rode about a dozen different Hunters, every single one went over fences in a Standing Martingale. Adjusted properly to only add pressure to the nose if needed. The ones I leased it was at the request of the owner or trainer . The ones I bought came going in them, occasionally they did save me some grief, they, the horses, were cheap and came with some issues, . The last Hunter I owned , who was not cheap or a project, didn’t need one so didn’t wear one after it became acceptable fashion wise to do so
I rode my late BILs rope and penning horses in tie downs at his request. The kind I described upthread. Adjusted to only come into play if the horse needed a reminder.
Anything is only as good as the hands and brain of the rider, even a plain, fat rubber snaffle…that is often too fat for the horses mouth.