You’re in luck, to this point, because your Veloster is relatively new — in model years — so that it features a fairly up-to-date engine control module (ECM). With that said, though, I think your trouble lies right there, the ECM. You will have to have this confirmed by having a diagnostic run by your dealer or a good shop with up-to-date software.
Now, without running the diagnostic myself, I can’t tell you what the exact problem might be but what I suspect has happened is that the ECM has gone into a loop and is refusing to communicate with any of the subsystems or sensors linked to the subsystems as well.
This means that when you put your foot on the accelerator instead of moving out nicely or even poorly — except very poorly — the subsystems connected to the pedal are in what you might call a large-sized quandary. It is asking the ECM for instructions and, since the ECM is essentially talking to itself and not taking any input, the engine defaults to a combination of limp-home mode (that’s the high-revving, low-speed alternative you mentioned — it’s there so you can get your Veloster either home or to a service area) and “deafness,” where it doesn’t hear from the subsystems or sensors and then just runs away.
My suggestion to you would be to drive the Veloster into a service area and have the technicians run the OBD-II diagnostic with the latest software if they have it. Then, using the codes that are likely to be thrown (likely to have the starting code structure p07XXX for transmission-related failures) will tell your technician where to look. I would further suggest that if you want the latest Hyundai-approved software so your codes would be far more precise in their error information so your techs will be able to find the problems quickly and fix the issues.
One last suggestion, for what it is worth. When was the last time you had the transmission fluid changed? It’s possible that the tranny, even though it’s a manual, could have very dirty transmission fluid and, as a result, it isn’t working correctly. I’d have the transmission fluid drained (not blown) out and swapped with new fluid and then see what happens. I would say, at this point, that this may be a long shot, but it is worth the $39.95 or so to have it changed. If things start to work correctly, you may have saved yourself hundreds. Let me know what your technicians find, would you? I like sharing with our readers so they can see what happens as the problems are remedied.