Red,
I'd double check that the fuel line is not restricted. Often times I've seen a little bit of debris in the fuel tank, typically grass or pine needle, that will converge at the fuel discharge causing a restriction while mowing. If none of this is happening, and you have good fuel flow, then the filter is fine. Most of them will not be full of fuel all the time when running. Sometimes you need to re-orient the filter to get it to hold more fuel more consistently.
The engine stalling when punching the gas is not that uncommon either. Remember, this isn't fuel injection. Just a butterfly that opens and the vacuum pulls more fuel into the carb. If things aren't perfect, she's going to choke and surge. A small vacuum leak can cause this. Dirt in the carb, carb out of adjustment, etc., could cause this too. Most tractors of that age you have to gradually increase the throttle to prevent choking, for any or all of these reasons. Also, I've been using Sea Foam - Deep Creep in carbs to clean them up, and it really works great. Take the air cleaner off and spray a bunch in the carb while not running. Also, disconnect the fuel line at the carb and spray a bunch through the fuel inlet. Let it sit for 15 or more minutes, then start her up and get it running at 3/4 throttle. Spray some more through the air intake, nearly choking the engine out, a few times. It really cleans out the carb and carbon in the engine.
The side panels actually do serve a good purpose. The design of the engine compartment utilizes the engine panels to help direct air through the engine in the correct direction. A REALLY good thing to do on those tractors is to remove all the sheet metal and blow out the cooling fins on the engine. They will plug with oil and gas and prevent the engine from cooling properly. Then get all the sheet metal back in place and get the side panels on.
What is the problem you're having with the side panels. I know one trick, especially with the older tractors like this, is to loosen the 4 grill mounting bolts at the frame, just enough to move the grill a little, slide the grill forward, install the side panels where you want them, bump the grill back so it is snug the way you want it, then tighten the grill mounting bolts. Believe it or not, this can make a huge difference in the way the panels fit. I've had many quiet line and newer Cub Cadets that people insisted the side panels were bent or would not fit, and in 10-minutes, I had them on the tractor and fitting like new. Sometimes this takes a few tries, but it works. Also remember, the side panels are designed to fit bottom side first then push in at the top. They'll never go on the other way, top first then bottom.
Bill