We have always put the saddle a bit higher up and slid it back down, first trying the saddle fit without a saddle pad, put the saddle on and check how it fits without a girth, on top of the horses back you should be able to put a hand in the canal on the wither and looking from tail-side should see light through the whole saddle canal.
Then put the pad and the saddle higher than needed, slide back till the saddle fits in a place itself and is level.
girth is always done up about the same holes on each side, if new girth or new horse we try slowly taking a few holes on each side and going back and forth, but for horses that are ridden with the same saddle and girth every day its already well known, what number of holes is put up on each side.
Shoulder always gets checked and girth is placed away from the elbow. If worried about saddle placement near the elbow and shoulder, sometimes you can lift the front legs of the horse, one by one, and stretch them out forwards, 1) gives a nice stretch before work after standing still for grooming, and 2) frees shoulders and all the skin under the saddle. Also lunging a bit before riding will help with saddle placement and tightening the belt.
most well fitted dressage saddles wont need anything else to hold them in place, jumping saddles always get a martingale with saddle holders ( not just a neck martingale, but it buckles up to the saddle).
However we do know a pony that is impossible to fit a saddle for, so many saddlies tried and all just slide almost on her neck in canter. she has very flat wither and is quite round, the tail piece scares her, she is very panicky about things going on her back/tail. even blankets can scare her sometimes..
Most important thing that a lot of people forget is to pull the saddle pad into the canal of the saddle before girthing it up, as if the saddle pad is too flat on the horse it will actually apply pressure on the wither and the horse will not be happy about it. However I have also seen very wide saddles used for horses with narrower back and higher withers causing the front of the saddle to actually lay on the wither... the person just claimed - nothing to do there... I was very sorry for the horse. I never use a saddle that doesnt fit well on a horse.