The Tamiya Basic Tool Set isn't inherently bad or anything. Their tooling designs are actually nice, but the basic set itself doesn't use exactly the highest grade materials, so they end up wearing down rather quickly. So it's entirely possible they've begun to dull. Cutting a piece is technically a quick action so we don't really register when cutting starts to slow down. However, when it does start to slow down you can start seeing more stress marks, which may be the cause here. It's also entirely possible you're cutting too slowly like you mentioned, though that can easily be remedied by just consciously trying to cut quicker. It definitely seems like you probably need to upgrade your tools though if you've been using them for a bit now. The God Hand is definitely expensive, so you don't necessarily have to go for that one (100% get it when you can afford it though. You'll never look back haha), but there are some cheaper mid-tier options that you'll immediately recognize as better.
https://www.gundamplanet.com/gundam-planet-premium-side-cutter.html
https://www.amazon.com/GSI-Creos-Nipper-Side-Cutter/dp/B00IFW9XB2
https://www.amazon.com/Tamiya-Sharp-Pointed-Side-Cutter/dp/B000J47Z4G
Any of those will serve you quite well without breaking the proverbial bank. You might want to try using that xacto knife you have. It may or may not do a better job. Basically what a good knife wants to feel like when cleaning up nubs is a slight resistance from the nub at first, but a very steady and smooth cut once the blade gets through it. If you feel too much resistance it's generally a sign that the knife you're using isn't sharp enough (For example, a lot of these knives are actually designed to go through simple stuff like balsa wood and paper). Tamiya actually has a top-notch one that's REALLY cheap you could get your hands on if you're interested in upgrading your knife as well.
Okay, so at first glance there's not an inherent difference between sanding sponges and paper. The top layer of the sponge is literally just a piece sandpaper material. However, it's in how they're used and perform where they differ. The most important thing for sanding is that whatever you're using to sand can both conform to the exact shape you need AND maintain firmness in the area making contact with the plastic. A piece of sandpaper can't do that. It can be shaped as you please, but it doesn't have enough firmness to maintain a perfect shape (and thus perfect contact with the piece you're sanding). That leads to an uneven sanding job, or even sometimes just not being able to get rid of a nub completely if said nub is in a weird place. There's also the added benefit of the sanding sponges just being outright easier to handle. Makes the process incredibly easy so you can focus more on the actual quality of your sanding job. At the end of the day actual sandpaper is better used for things like the huge flat surfaces of a wooden structure.
Since you mentioned the proper grit range this knight is inclined to say it's largely an issue not enough hand pressure pressing the paper into the piece, and the paper used just not being good for this sort of project. Honestly, what I'd recommend is trying the sanding sponge-stick sets below. They're ultra cheap but high quality (think you can buy them on Amazon and HLJ if those are more to your liking too). They also cover the ranges you need perfectly.
https://www.amiami.com/eng/detail/?gcode=TOL-00001456
https://www.amiami.com/eng/detail/?gcode=TOL-00001450
It seems silly to say since sanding is such a simple motion (rub the thing into this thing!) but it's actually got some nuance to it you eventually get used to. What helps a lot though is having a good sanding tool. Once you have a good sanding tool your hands adjust to the amount of pressure you need to apply rather quickly. So I'd recommend picking up some sanding sponges and then practice removing marks on a model's runner or something. You should get used to them and notice a huge difference in your workflow.
I believe the Xuron are the better of the two.
There is also the MR. Hobby Nipper https://www.amazon.com/GSI-Creos-Nipper-Side-Cutter/dp/B00IFW9XB2
Personally I prefer nippers with springs