What sort of injury did Diesel have? What does your lameness vet say? Either a horse is sound, or they are not. So it doesn't make sense that he would be sound enough to show, but not sound enough to barrel race. He needs to be sound to show too, as that requires excellent collection. Plus, a judge would spot him if he's not sound. So if he doesn't regain soundness, he needs to be retired.
Granted you posted only one picture of your barrel racing on Diesel, but your position is anything but ideal. You're looking down, you're using your outside hand instead of inside hand to rein him around the barrel, and it appears like you are ready to smack him with the end of your reins before you've even fully completed the turn. Not saying any of that to be harsh, but that's what I see in the picture.
Did Diesel seem to enjoy barrels? Some horses like it and some don't.
So Daisey ..... doesn't stop ...... and doesn't turn....... yet you probably asked her for full speed at the 2 barrel races you took her to. Exactly what sort of barrel training have you put her through? How long did you train her on the pattern before competing on her? It really sounds like you pushed her too fast too soon, and she's very confused. Hence why she's "lashing out" and getting completely uncontrollable.
I agree that Daisy is standing rather "camped out" in the photos you posted. Have you had her checked by a vet or chiro? That can also contribute to a horse getting out of control on the barrel pattern, if she is in pain.
My favorite thing to do is barrel race. However, I like to go to the local showing events too. I like my barrel horses to be well-rounded individuals. They know precisely when it's time to turn-and-burn, and when it is time to be show pretty and calm.
At one of our local shows, I entered my horse Red in every single event. If my memory serves me correctly, we took 3rd place in Halter, 3rd place in Western Pleasure, 4th place in Reining, 2nd place in Competitive Trail, 2nd place in Pole Bending, 1st place in Ring Race (speed event), and narrowly missed placing in barrel racing. Local show, yes, but there were about 12 to 15 people in each event. Not too bad for his first year of showing/gaming, and being such a well-rounded individual. But I've been training him for barrels/poles for over a year, and working on general riding during that time too. He was brought along slow and correct, so he's got a sound mind for barrel racing.
One horse is not going to be a world champion in every event. But they may excel in one event, and do "okay" in others.
If you like to do all the events, then train your horse for all the events. That's what I do. But make sure you don't rush into the barrel racing, because it sounds like you are taking it too quickly with Daisey, and she's blowing up on you. Check out some barrel tips in the other forum: Barrel Racing Exercises and Drills.