I'm no guru (other please correct me if i am wrong in any aspect).
Even on the non-running ones Your left foot sometimes leaves the ground on yyour forehand and i don't think that is giving your proper balance. I'd imagine if you keep it on the ground and really push off that back foot and rotate you'll be able to hit better. You were doing it correctly on some stokes. I'd just keep both feet on ground.
Baby steps right before you get to ball, especially on the move you have to take those baby steps to adjust. You never know 100% exactly where that ball is going to be and where your body is. Those baby steps allow you to properly adjust.
These were two tips I have for hitting on the move.
(1) try turning your body while on the move, i'm sure you have tried those karaoke warm up steps its similar, but I don't believe most rec players will actually rotate their body as they are running to prepare for impact. You might start taking your racket back while moving but i'm actually talking about rotating your core.
(2) don't be afraid of taking things early. Work on your timing and see if you can interest the path of the ball. Rather than move parallel with the baseline take a more direct path to the ball as it should be a shorter distance (depends on shot) ex. player hits cross court shot to my forehand, rather than move parallel with the base line i take a step in and cut off the ball. I take time away from the opponent and its a shorter path for me. Of course this doesn't work for down the line shots as distance would be the same. Just don't be afraid to take the ball early.
Hopefully those helped =p. Good luck!
Head Youtek IG Ltd. Edition Prestige 348g 10 pts HL (x2) | Big Hitter Black Zone 1.20mm @ 40 lbs | Leather Grip + Supergrap | Rubber Band Dampner