What would be the best thing to do in this situation? (As in upgrading and upgrading to what?)
What to do is up to you. I would do the following:
Use appropriate camera settings. See below for details.
Seek a camera that lacks an AA filter (or has one with reduced strength). I prefer mirrorless cameras. There are many options from Nikon, Sony, FujiFilm, etc.
Look at different lenses. Primes and constant aperture zooms usually produce good results.
Electronic adapters from EF to various mounts are available, so if you have other EF/EF-S lenses that you like, you can still use them if you decide to switch camera systems.
Is it true that full frame sensors have better quality than cropped sensors?
For sensors with the same resolution (megapixels), full-frame sensors are able to produce images that appear sharper because the sensels are larger. The difference is approximately proportional to the crop factor. Suppose a lens that resolves about 65 lp/mm is suitable on a given full-frame camera. The crop-sensor camera would require 104 lp/mm to produce an image that appears just as sharp.
Similarly, low-light performance of the full frame sensor (again, same resolution/megapixels) would be improved because of larger sensels, proportional to the crop-factor squared (~2.56 for Canon, ~2.25 for other brands). This is the approximately one-stop difference in ISO performance that people often refer to when comparing full frame with crop sensor.
However, if the pixel density of the sensors is the same (same sensel size, different resolution/megapixels), there are no advantages as far as sharpness and low-light performance are concerned. (Assuming no other technological differences between the sensors.)
There are also issues with corner and edge performance that may be considered. (Sharpness, coma, light fall off, etc.) Depending on the lens, full-frame sensors may capture edge and corner effects that wouldn't be captured by crop sensors. Using the same lens, image quality on crop sensor would be more even across the frame than with full frame. If this is desired, crop sensor may be considered "better". But some people like the edge effects for "character", so for them, full frame may be considered "better".
When taking a landscape photo with a deep depth of view, the image is not sharp, especially the further away it is.
Optimal lens performance is usually with the aperture stopped down 1-2 stops. On your zoom lens, that would be ~F5.6-8 at the short end, and F8-11 at the long end. However, when apertures are smaller than F8 (larger F numbers), diffraction tends to introduce noticeable softness.
To determine the lens parameters that would result in the "same" captured image on a crop-sensor camera (vs full-frame), divide the focal length and aperture by the crop factor. Suppose you want results that look like 50mm F11 on full frame, you would need a 31.25mm F6.875 lens on crop sensor. Since the aperture is wider, you avoid diffraction that typically accompanies F11. Crop sensor may be considered "better" in this case because it allows the use of wider apertures to obtain the same DOF.
You may have better results with different lenses. Primes are usually considered good choices. However...
I am currently using a Canon EOS 4000D with its 18-50mm kit lens.
Your camera has an antialiasing filter that intentionally blurs the image to prevent moiré, which is typically not a significant issue in scenes that lack artificial patterns, such as fabric textures.
Here is a sample image showing the difference between cameras with and without AA filter using the exact same lens (EF 40/2.8 STM) and exposure settings. The difference can be seen even at reduced resolution in the thumbnail. (So it's not necessarily true that sharpness doesn't matter when reducing image sizes for the web.)