Well if she can flat walk she can running walk. What you could be getting is a horse that is totally confused in her gaits so trying to do a couple different things at the same time, that usually leads to crossfiring and a rough ride. Now, if she is gaited you want to stop trotting her and just work on her gait as that wont help. The other thing is gaiting takes building up the proper muscles, if those muscles are not strong it leads to going into/out of gait.
Show one of your friends the video you posted and point out what a proper 4 beat gait looks like. Hop on your mare and dog walk, that's a slow walk, have your friend watch and see what her feet are doing. It should look just like the flat walk but slower, front foot should barely get out of the way before her back foot lands. Also watch to see if the back foot lands behind, same place or in front of where her front foot was. Walkers should put their back foot in front of where their front foot landed, this is called overstep.
If it looks like she is gaiting then it's time to start working on her gait. First, dog walk her until you both are sick of it then dog walk some more. This is to build up the proper muscles for gaiting, gaiting is work and they need to be conditioned to do it. While working her dog walk I would do two things. Buy a couple books on training walkers to read and look for a walker trainer that can give you some lessons on one that is set in their gait already. I also suggest sending her to a walker trainer for a month or so as neither of you know what a gait is by the sounds of it. You can do it on your own but it will take a lot longer and you may never getting her 100% gaiting right.