Although we're experienced showing people (cobs, M&Ms, show Ponies, Show hunter ponies and workers) we don't know much about senior showing I'm afraid.
Therefore you may be best downloading the rule book from the Senior Showing and Dressage and joining up ahead of Windsor. It seems that you may need an age badge for your horse to wear on it's bridle and the rule book explains everything about tack and your own turn out (no earrings, spurs etc). You may want to call them for more info..
I believe you'd show like a hack ie everybody in on right rein at walk, then trot, then canter usually on both reins. Definitely not galloping at any stage in Seniors..
Then you get called in but not as a "pull" (in order of preference) and will have to do an individual show. I presume you will be stripped but again check with the society.
Sorry can't be more definite. I'll know more after watching the senior classes at BSPS Area 16 Show (Trinity Park, Ipswich) on 8th April as they are the same classes as Royal Windsor..
Hope it all works out for you. How old is yours?