Basically the best way around this other than the obligatory mirror shot would be to use the zoom function on a point and shoot camera (non-digital zoom, you want optical zoom) and zoom it in as much as possible, generally around 4X should be enough. The problem created from this would be that a self-portrait can no longer be done from arms length and would have to be taken by someone else, or use the timer feature and set the camera at a distance.
What is creating this effect which you do not like is barrel distortion created from a wide angle lens / wide angle range, this is fairly common in wide angle lenses, no limited to point and shoots, even found in SLR lenses (even the high end ones) depending on how wide the lens is (how much crap you can fit into the picture).
Here is an example of this effect in the extreme on a wide angle:(Source)
Generally when you are going wide on a point and shoot, it is great for landscapes, or anytime when you need to fit a lot of things in the image, but things will get distorted when you get up and close.
What you usually see with most portraits are lens ranges of about 50mm to 100mm+, if you want to get into this range as stated 4X+ optical zoom will be sufficient.
*EDIT* there are ways to get around this if you do not have a camera that can zoom in optically, such as photoshop or adobe lightroom.