That is the TWH "camel" walk or swingy walk. The horse you are looking at probably has a big overstride. My mare does not have good shoulder conformation and so hers turns into "thumpy-walk" which throws me back and forward in the saddle about 4 inches, particularly downhill. It is really uncomfortable and I would not go on a trail ride with others because of it. I can control it by collecting her into a slow walk, or speed her up into a flatwalk (which is a wonderful gait). But if I leave the reins loose I get a walk I dont like that is physically hard on me.
My mare has other faults too, but she is so gentle and forgiving and safe and reliable that I can forgive her many more faults. If you are 59 and looking for a horse the TWH breed is an excellent place to start looking for the right one as not getting hurt and feeling safe, and being safe are so important. There are probably a higher proportion of tolerant placid minded people-minded horses in the TWH breed than many others. Mine tolerates and works with absolute beginners, was gun/whip safe without ever having been trained first and is just so tolerant and forgiving under saddle.
I would ride that horse you are interested in for a few hours, on a trail ride, particularly with others if that is what you want to do with it because if you dont like that walk you are NOT going to be able to fix it - it is how the horse is made. On the other hand, like the other poster said, they have MANY walks and gaits that they can do, but if that is the predominant walk gait for that horse you dont want to be fighting what is the default way of going. That horse will probably flat walk, runwalk and rack well and also may have a beautiful soft "to-die-for" canter because of the over-stride, but the down side can be a swingy ordinary walk.
If a horse is good to handle and nice on the ground does NOT mean it will be under saddle, dont mix that when horse shopping. Try the horse out on trail rides if you can. If it is difficult for you to sell a horse that does not suit you then be careful and spend the time and money it takes to get it right in the first place. DO NOT COMPROMISE on temperament under saddle and safety. There are very reliable horses out there, particularly in the TWH breed. I believe truly beginner safe horses are bred, not trained (I dont mean that they dont need training). My mare has been beginner safe since green broke.
Good luck horse hunting.