I have to disagree about going out and buying a new saddle…
I’ve never had a lot of money, and have had to make do with the school horse’s saddles, whatever saddle came with the horse, hand-me-downs, etc…
You CAN develop a good seat as long as the saddle is a relatively ok size for you (and I’m assuming your trainer would be able to tell if the saddle was much too small for you or other such glaring problem.)
A chair seat is usually the result of weak thigh and core (abdominal) muscles and/or sitting incorrectly on the seatbones. The age-old fixes apply: lunge lessons, exercises to develop the ‘independent seat’, riding without stirrups, developing inner thigh and core muscles with exercise off the horse.
A lot of riders try to achieve the ‘dressage seat’ by letting down their stirrups too long, and tucking their pelvis under too much… the leg tends to turn outward and the pelvis tucks under and the lower back rounds, the abdominals are collapsed… the rider ends up sitting not squarely on the seatbones but farther back on their butt and the leg just slides forward, and there you have your chair seat. If you rode hunt seat before, try to remember how nice it felt to have your heels down, your back slightly arched and your knee and inner leg securely against the horse.