OP, a lot of draft horses are VERY laid back and easy going. I'd first find a trainer/riding instructor to help you shop for a horse, that way, you have an educated and experienced eye that can catch things that you might not be able to. A draft or draft cross is a very good option, one with a short back especially. A horse with a long back will have too much stress put on its back, with any rider, not just a plus sized rider. Even a halflinger or fjord (they're considered draft ponies although a lot of them can make 14.2h-15h) would work and they'd be closer to the ground if you were worried about height.
Being a new rider, you should look for a horse over 10, maybe about 12-15y/o that has been there done that, beginner/husband safe and can be a confidence builder for you.
the common myth that a draft eats an incredible amount is also what I believed till I owned one! Just to further clarify for others reading this...they do drink a lot more water due to their body surface area but their metabolism tends to be lower so all the feed per pound of body weight tables really do not apply
Op..good luck in your search for a horse it is an exciting but nerve racking time..but so worth it... the first time your horse nickers as you approach will melt your heart!