I have shot at several Appleseed events and I am struggling with exactly the same issue you are. I shot 22 lr rifle way back in Junior High School (think 1960s) and just started taking up rifle shooting again a couple of years ago. At Appleseed I have been able to fairly easily qualify as Rifleman each time, with scores generally in the 225-235 range, but offhand has far and away been my worst stage, and I would really like to improve my performance standing.
For myself, I think part of the problem is psychological. I remember distinctly the first time I tried to shoot standing when I was a skinny, weak teenager. I felt as if I was lucky to hit the wall at the far end of the range at 50 feet. I got the impression right then that I stunk shooting offhand, and that low expectation has sort of stuck with me.
As for shooting offhand at Appleseed I am going to throw out some thoughts that seem to be making a bit of a difference for me. Keep in mind these are not coming from some great offhand shooter, but from someone who has been trying to address exactly the same frustrations you are.
First, I have become convinced from talking with very good rifle target shooters that the two key elements for becoming a decent offhand shooter are discipline and practice. You need a lot of practice to refine your stance and trigger technique and to strengthen the muscles required.
Discipline is obviously required to make yourself do all that practice, but you also must have the discipline to simply not accept a bad shot. On stage 1 of the Appleseed AQT you have plenty of time so resist the temptation to accept a poor shot. I tend to hate the standing stage so much that I sometimes find myself rushing just to get the misery over with. This tendency just becomes worse and worse as you get tired. I tend to keep my standing practice sessions relatively short to avoid the bad habits that come with fatigue.
Stance is critical, and I have become convinced from watching other shooters and studying photos of top high power shooters that the same stance does not work for everybody. In any shooting posture, you want to maximize balance which maximizes skeletal support and minimizes muscular contraction and effort. Balancing requires taking into account the center of gravity of both you and the rifle. The rifle will place the combined center of gravity in front of your spinal column and somewhat towards the target. If you look at photos of many top shooters, they compensate by leaning their trunk somewhat back of their hips and away from the target. One champion rimfire shooter told me that she is always concentrating on eliminating unnecessary muscular tension during a stage. It is natural to start to "tense up" during a stage, perhaps even more so if you perceive you are doing well and starting to mentally tell yourself not to blow it.
I find that if the rules allow, as they do in Appleseed, rifle support is greatly enhanced by bringing the support elbow in against the body near the hip. If you look at photos of top high power shooters, they all seem to do this. It is also important to keep the head upright. On average the head weighs 8 pounds or more, and if you drop it way forward and down onto the rifle, you will reduce your skeletal support and balance. So bring the rifle butt up as needed to get the head upright.
Depending on your arm length and the target height, bracing your support elbow in against your body and keeping your head upright might require you to experiment with the configuration of your support arm hand. Some shooters will need to support the rifle with extended fingers and thumb or a lightly closed fist to get the sights on target. Some use a "lobster claw" or "reverse lobster claw" configuration and some support support the rifle on their open palm with all of the fingers and thumb on one side of the stock. Experiment to find what works best for you.
I have tried shooting with and without a sling for the offhand stage. So far, I seem to get better results with a sling. I use a leather 1907 type loop sling for sitting and prone, and switch to a USGI sling in the hasty configuration for standing. I have either a second spring steel J clip at the front end of the sling, or quick release button sling swivels at both ends, depending on the rifle I am using. This allows me to rapidly attach and detach the sling. There are two small tricks I have found that make the hasty sling work a bit better. One is when attaching the sling at the front of the stock, give it a half-twist clockwise before putting your arm through. This allows the sling to lie more smoothly against the wrist of your support hand. And after putting your support arm through the sling and getting it as high as possible, before mounting the butt of the rifle to your shoulder, pull the rifle down toward the ground to tension the sling. I bought an inexpensive leather shooting glove from Creedmoor which provides additional padding for the sling. If you have not tried shooting with a glove, you might consider doing so.
I would like to shoot with iron sights, but my eyes are simply no longer good enough. I need a scope especially to see those little "400 yard" scaled silhouettes. I have a fixed magnification 4X scope on one Ruger 10/22 and a 3-9X variable magnification on the other. For the offhand stage, with the variable magnification scope I keep it dialed down to 3X. I find that anything more than 3 or 4X magnifies my "wobble" too much which tends to increase my general anxiety and tendency to try to snatch the shot.
Although I avoid the tendency to snatch the shot, I do try to time the trigger break so that it coincides with my natural respiratory pause. I try to achieve a NPOA such that my scope is centered at end expiration. Unless you are very unusual, or doing something unusual, your sights will rise as you inhale and fall as you exhale, so there will be a natural up and down motion as you breathe. I have found that if I time my trigger press so that the break occurs just slightly before the sights drop to center, this up and down movement tends to reduce any side to side wobble. I used to find myself moderating my degree of exhalation to adjust the sight picture, but for me this was not consistent. So now I try to develop a cadence that allow the shot to break just as the sights drop down onto the center of the silhouette. If on the other hand, I exhale completely and then pause to try to center the sight picture, I start to get more of the dreaded side to side, or figure eight wobble.
(I was almost 10 years younger...)