I appologize but at 120+ lbs, you are
too large for a shetland pony. As far as I know, shetlands can only be up to 11hh, or 11.2 hh for American Shetlands (may be wrong on that part?). 11 hh would be 44 inches, which is only about 3'6 at the withers. That would come a full TWO FEET shorter than you, so the pony's back would probably come up to about the bottom of your rib cage.
Our shetland gelding is 44 inches, pretty much the tallest a shetland can be. He weighs about 420 lbs, and probably ought to weigh more like 400 lbs (he gets fat on air, I swear o.o). Using the 20% rule, that would mean he can carry about 80 lbs when in good shape, including the saddle. Without a saddle I wouldn't want him to be carrying more than 60 or 70 pounds because the weight will be unevenly distributed and placed in one focal point, which increases wear. Using 80 lbs as our guideline though, that leaves you at 40 lbs too heavy. The minimum weight of a pony that you should be riding is one that is roughly 600 lbs when not overweight- which puts you at about the correct size for 12.2-13hh pony. For that you would need a pony of the icelandic, welsh, caspian, or possibly POA origin.
I am 4'11 and I weight about 90 pounds on my 'heavier' days. I absolutely refuse to ride anything shorter than 11.3 or so, even for a five minute schooling session- and even then I feel far too heavy and I would never ride a pony that wasn't in shape that was this size. There is too much risk, even if it is only for a minute on scarce occasions.
Here are a few pictures to give you an idea of the height you're talking about.
This is the 44 inch tall shetland. The girls driving him are 5'3 and 5'5, shorter than you and much too tall to ride him, but perfectly fine to drive him.
This pony is 11.2 hh, the absolute shortest I would EVER sit on as a 4'11 teenager. Her rider is 60 lbs and 4'2 which is perfect. Now imagine where your legs would be on this pony.
This mare is 13.2, almost 13.3hh. His rider (in the back) is 5'7, a bit taller than you. Obviously too tall, and probably a little too heavy as well. He also looks a bit silly, but for the purpose of a short walk, he'd be just fine.
(btw, I do not condone the use of a mechanical hackamore, but it is not my horse)
Lastly, this filly is obviously only a year old any not old enough to be ridden, but here is her height of 12.2hh in comparison to me. As you can see, her withers are a bit below my shoulders. If she were to be a mature horse at this size, I would be comfortable riding her but I would not let my friend, who is 5'6 and 140 lbs, ride her. This filly's withers only reach about half way up her back.
I hope this helped. If I were you, I'd either just buy a mini and never ride it, but rather have it trained to pull a cart like my mini does, or I would wait and buy a pony /horse that you could ride consistantly, that stood at at least 13hh, better at 13.2+.