The picture distortion is called keystone. Unsurprisingly then, the fix for it is called keystone correction. There are two types.
Electronic keystone is simple and widely used. The picture is electronically shrunk and distorted within the boundaries of the imaging chip to compensate for the optical distortion caused by the projection angle. However, there are two problems: First, by remapping the image it generates a lot of image artefacts so you get staircasing on what should be straight edges (jaggies), and a general loss of resolution. It affects everything, but you'll notice it most with onscreen menus and fine detail.
The second issue is one that folk rarely consider. Reducing the image size on the chip has the effect of messing up the projection distance. The projector no longer follows the manufacturer's throw ratio guide. The projection distance required to fill the screen becomes longer. When you buy your screen based on the distance from the shed, then find that the corrected image won't fill it like you planned, but all the maths looks correct, then this is the reason why.
There are a couple of alternatives to electronic keystone correction. The first is Optical Lens Shift. The second is short-throw and ultra-short-throw projection.
Short-throw is easiest to understand. The projector has a lens that allows for a relatively large image at a relatively short throw distance. This is quite common on gaming and entertainment projectors. The image width equals the projection distance, and the projector sits below the bottom line of the projected image. Some can get closer than that. The trade-off is the loss of a zoom function. This means to make a bigger or smaller image requires moving the projector further or closer to the screen.
The other solution is Optical Lens Shift.This uses a projection lens that can move left-right and up-down which has the effect of shifting the image in the corresponding direction. The advantage with lens shift is you retain zoom and don't introduce any image artefacts. The catch is cost and a limit on how far the image can be shifted off-axis. The norm is 1/3 to 1/2 of the image size in either axis, but not maximums at the same time. For example, a 6ft-wide image could be shifted up to 2 or 3 feet left or right.
The previous poster is correct about those cheap LED projectors. They sound great on paper, but the adverts are written in a way that makes them out be to be far higher specified than they are in reality.
In this situation, a used projector would be a far safer bet.
There are plenty of really good 720p and 1080p used projectors going for peanuts. Mostly it's down to lamp replacement costs.