Using a projector with Dell XPS 13 at 3200x1800 resolution

There are 2 separate problems involved here, and no perfect solution -- but you can get better than what you're seeing.  But before I get into those, I'll note upfront that even apart from the problems I'm about to describe, getting away from VGA alone should improve things because analog video looks noticeably worse than digital, especially at 1080p.  I'll also note that some projectors can accept higher resolutions than they can "truly" display.  For example, lots of projectors have a "native" 1280x800 resolution but will accept a 1080p signal and then just scale it appropriately.  All else being equal, a projector whose native resolution is 1920x1080 will look better, but those are of course more expensive, again all else being equal.

Anyhow, the first problem is that your built-in LCD will always look blurrier whenever it's not running at its native resolution.  1600x900 might work reasonably well since it's exactly half the native resolution in each dimension, but even that isn't totally perfect because Windows doesn't just double up the pixels that way.

The second problem is that you may have noticed that on your built-in panel running at 3200x1800 resolution, you have display scaling enabled -- probably 150-200%, because 3200x1800 on a 13" display would result in microscopic text at 100% scaling.  The trouble is that when using multiple displays with significantly different resolutions and/or pixel densities, Windows can't just use a 150-200% scale factor across the board.  That would look huge on a 24" 1080p display, for example, and even worse on a projector.  However, Windows only really optimizes its scaling for a single scale factor, and then it basically just scales a pre-rendered image up or down to make elements appear a reasonable physical size on the other displays.  This does not always look great, and it's why a 1080p external display might look fine when used on its own (since Windows is optimizing for it) but can look blurrier if you also have a 3200x1800 display active.  This has been getting better over successive releases of Windows 10, but it's still not perfect.

So you have two options here:

#1: Set your built-in display to a lower resolution like 1600x900 and set a 100% scale factor, then log off and back on for the scale factor change to take full effect.  Windows will then be using a 100% scale factor on both your built-in and projector displays, so the second problem goes away entirely.  Of course your internal LCD will still be a bit blurrier, but that's how it is.

#2: If you want to keep a high resolution on your internal panel, then the second problem exists because Windows optimizes its scaling for whichever display is set as PRIMARY, so you could instead set your projector display as primary (and then log off and back on so Windows changes its "native" scale factor to the setting appropriate for the projector), and then the projector display will look its best.  However, your built-in display may look a bit worse than normal, despite still using 3200x1800 resolution, because you've essentially just flipped the second problem around.  The other downside to this is that your Start button, taskbar, and desktop icons will now appear on the projector display, which typically isn't desirable.

So essentially, you can make the projector look better, but only by accepting some blurriness on your built-in panel, and you can have blurriness caused either by running a lower resolution or running the native resolution and having scaling imperfections.  The bottom line is that running displays with vastly different resolutions simultaneously doesn't have a great solution, and that's due to a huge number of reasons about how graphical interfaces and elements are developed and rendered, how Windows has changed how this works over the years to support new display technologies, and the fact that applications don't always add support for the new methods, so you can get very mixed results even from application to application.

But whatever you do, stop using VGA! :)