I don't know about that particular displaylink-adapter, it does look pretty basic but at least in theory OK for 1080p, but I'd say in general DisplayLink can do great job at delivering convincing near-native image quality, if given enough bandwidth. Especially regular text and stuff should look near-indistinguishible between the one connected to the M2's display controller and the one connected via the adapter.
So the question is, does one of the monitors look different to you? Or are you just unsatisfied with the "pixeliness" of both of them in general? Maybe a photo could help us here.
Because 1080p on 24" is pretty bad regarding pixel density, and MacOS doesn't really bother with much subpixel smoothing like Windows does, so text and such will not look that great regardless. But it depends on how picky you are. It might be that you want to go with higher pixel density monitors and that will cost a bit.
A dual display solution for two 4K-monitors with the Air can in theory work with a proper DisplayLink adapter. You'd need something like
this, an adapter than can truly do 4K @ 60Hz. I would
only connect one monitor through it however, since each monitor connected through DisplayLink eats a lot of CPU resources as the driver essentially has to live-encode down a screen's worth of content into a video stream and send it out. This effect is also excerbated by the high resolution we're targeting here. The other one I'd connect via USB-C -> DisplayPort/HDMI and thus use the M2's built-in controller.
If you can live with one monitor being low-res, then obviously plug that one into the DisplayLink-adapter, and maybe you don't need to buy a new one of those either.
There is a lot to think about when wanting to go high-end with monitors on a Macbook Pro, however. Especially if you think you're going to end up being picky with the results. I want to say that the M2 Air is not the ideal macbook for this.
Here is also some good reading to start with before buying any monitor for a Mac.
