Yes, you can take your next turn while still under the effect of Grovel, Cower, and Beg. It doesn’t make a huge amount of difference, though; you can’t benefit from your own groveling. Only your allies gain advantage, not you, so while you could do something on your turn that triggers readied actions by your allies, that would be a pretty unusual set of circumstances.
As far as going prone, no, the action doesn’t specify that you go prone to do it. Prone is a fairly specific position and the condition applies a moderately large penalty that I wouldn’t apply unless it’s specifically called for or very obviously applies (such as if an attack comes when you had specifically just laid down for a nap).
If you need arguments to present to your DM, the main thing I would note is that there’s a significant difference between kneeling or crouching and being prone (i.e. flat on your back or front). Getting up from prone generally involves a lot of maneuvering to get your feet back under you, while if you take a knee, you can spring right back up and straight into motion at almost any time (depending on the condition of your knees). As generally understood, cowering doesn’t mean planking or something — you aren’t stretching out full length on the ground, you’re crouching and trying to be as small as possible.