The singing and laughing advice you've received is spot on! Here's part of the why: you can't hold your breathe while you're singing. Laughing makes your muscles jiggle.
In your pictures, she seems to be a sensible mare. My impression is that she's a bit hyper-tuned to you. What I mean by that is she's inclined to take her behavior cues from you. She's got her ears back and is quite tense.
She's also a bit peeved about the kick strap.
If she's wanting to scoot out from under you and run, you could try harnessing her up and cantering her while ground driving. Is she wanting to run as soon as you hook up? After you start walking? From the trot?
Also, when you hooked her up at the end of September and she did alright wasn't that with a different cart? Is it possible that when you hooked to your repaired cart that the balance or draft or something has changed?
Honestly, that last tense--and a bit crabby--posture in the last photo really makes me think she's peeved about something. As if she's saying, "I don't know why you don't trust ME anymore, I'm a good girl.
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Edited to add: I did go back and look at the pics again close-up (on my cell.) It seems you have it adjusted tighter than the hip straps for your breaching. It looks as if the hip straps are floating on top of her fluffies while the kick strap is pressed down right to skin. Maybe loosen her up one hole, at least on one side and see how she does?