Isn't falling the worst feeling???? You know you're going, seems like in slow motion, but you can't do anything about it.

I have three different canes -- all stuck away in the closet now -- but I wish I had put my vanity aside back then and used a rollator instead, especially one with a seat! As others have said, a cane helps somewhat with balance, but if it gets worse and like you mentioned, your leg just gives out, I don't think a cane would help very much.
I remember meeting the blacktop in a Sears parking lot, up close and personal -- tripping up a step (foot drop) to Radio Shack, landing on my knees, stomach.

After thinking a walker/rollator would be too embarrassing, it wasn't too cool lying on the ground either with holes in the knees of the pants and blood oozing out.

Also as time went on and my balance got worse, trying to walk through a big "open" section of the mall, brought on the vertigo. Walking with a bunch of packages while trying to hang on a cane for dear life isn't very much fun either. What I would have done for a place to sit (rollator).
So.....my advice.....for what it's worth.....would be try a cheap cane and see how it goes, but keep that rollator in mind. You'll be able to use it longer. In the big box stores, use a scooter they keep there. It's hard at first, but fun once you get used to it. My grands always wanted to sit on my lap and "drive" if they were with me. I took out a couple end displays before I perfected my "driving skills". Use whatever you have to in order to remain mobile....it's really the pitts staying home all the time now.
Take care and be careful!