Wow, does sound like clippers are not adjusted correctly. Tightening the tension usually helps.
I wonder how dirty the old horse is? We wash the horse before clipping. Clean hair/skin lets the blades do a smoother hair cutting job and blade sharpness lasts much longer. We usually can clip an entire large horse with one set of blades, with a nice looking result. Clipping dirty horses dulls the blades fast, you will need several sets to get horse finished. Then it costs to have them all sharpened again!
I have several sets of the large Oster/Sunbeam clippers, all are old, work well, have many hours on them, still work fine. The guy who sharpens my blades will also service clippers. Mine have been repaired a few times over the years, new cords, that square brass piece by the arms was worn and got replaced, noisy got fixed. I had him look at some used clippers I got, before using them. He is quite reasonably priced, with quick turn around returning blades and clippers. Perhaps there is a local guy near you who can do that for you. Mine does mail in work too. If you want to contact him, he is Precision Sharpening, 517-521-4371, Webberville, MI.
I have several clippers to allow cooling time during the massive clipping sessions we sometimes have. Running hot blades makes the blades lose their “temper” which is the hardening in the metal. Such blades then are not going to stay sharp, cut badly. Also hot blades will have horse dancing as he gets burned. Hot to the touch of finger, means it is time to let the blades cool off. Better to change clippers to a cool blade set, rest the hot ones back to cool. Saves your blades sharpness, keep the horse happy, standing well for clipping. We do 4-5 horses in a body clipping session, swapping the clippers as we go along.
Clippers have also done show cattle clipping, sheared show lambs, with excellent results. Can’t have lines or uneven cutting on those show animals, indicates lack of care presenting them.