I'm just trying to describe it for a short story I'm writing? I figure that one would describe the way someone moves in crutches differently than walking, and the only word that comes to mind is "swing" and that sounds a little odd in my opinion.
Two words to describe a broken gait are "to limp" and "to hobble".
"Hobble" suggests one is walking unevenly due to injury or pain. Limp mean to walk with difficulty due to damage, weakness or stiffness. So either would be correct.
You can say "He is limping" or "He is walking with a limp". You can also say "He is hobbling on crutches", but not "He is walking with a hobble" Check a dictionary for more usage notes.
The simplest alternative to "use crutches" or "walk with crutches" is "to crutch".
A onelegged sailor crutched himself round MacConnell's corner, skirting Rabaiotti's icecream car, and jerked himself up Eccles street.
James Joyce, "Ulysses"
He was recorded crutching down a hallway at Arrowhead Stadium Sunday.
Jacob Klinger, "Steelers short 3 offensive, 2 defensive starters," 18 Oct. 2017
After a quick “how-to” from the nurse, I quickly crutched my way out the front doors ready to start the day.
Suraj Sehgal, "What Crutches Taught me about life," 08/24/2016
I crutched into the neighborhood barbershop and got myself a trim, and I picked up a footlong Cold Cut Combo from Subway.
Dad's Daytime Diary, "Walker vs. Crutches? No contest," September 18, 2010
I crutched down the long ramp and onto the plane.
Re: Airline travel with crutches- what to expect?
Most of the amputees one sees crutching around or being wheeled around have a prosthesis hidden away in some dark corner.
--Lenor Madruga, "One Step at a Time"
There's no room for it when you're tall and foreign and crutching around China.
Yin-Yang: American Perspectives on Living in China - Page 121, edited by Alice Renouf, Mary Beth Ryan-Maher
Crutch as a verb can also mean to support as if on crutches, or to prop up. I think that usage is exceedingly rare (at least in AmE) so you shouldn't need to worry about it.