Here's my two cents worth... If the little rails on the toddler bed aren't preventing her from falling out, then she's not ready for the toddler bed. I would just put the mattress right on the floor--forget the frame for now--and see how that goes.
Or, do you have a playpen or pack-and-play? She could maybe sleep in that for the interim, and you could try the toddler bed again in a few months.
If you do want to go with different side rails, PLEASE check with the bed's manufacturer to see what, if any, other bed rails you can use with that particular bed. To answer your question about getting in and out of bed if there are longer side rails--I have never seen bed rails that go the full length of the bed. (Doesn't mean they don't exist.) The ones I've seen go 1/3 to 1/2 down the bed, leaving the bottom portion of the bed open to get off and on. If she is moving around so much that she is "leaving" the railed portion of the bed, I would say she is not ready for that bed.
My son is 18 months and moves all over his crib. Like I can go check on him every 15 minutes and he will be in a new position. Half the time he actually has his feet elevated and resting on the side of the crib, LOL.
Lastly, I'm confused about one point in your post. Did you move her from the crib to the toddler bed while she was sleeping? If so, I would put her in the toddler bed while she is awake. She needs to see where she is going to sleep and get a feel for the lack of "sides" and how much room she has. I would never move her while she was sleeping.