Kind of off topic, but PSP 7.04 (silly me, never remember to upgrade!) does have a problem saving 32-bit PNGs. I happened to have an old clunker on hand, MS Image Composer, which can convert 32-bit TGAs into them (which I need for another game).
On-topic, one thing I would try is looking for photos online of wolves, bears, etc. Take a snippet and use PSP's ability to turn a selection into a repeating pattern (found in the "Selections" menu). Make sure the texture of the fur is easily visble, otherwise it will come out looking less like fur and more like long underwear

You can then take that tiled fur texture and piece it together in a new layer (Edit --> Paste --> As New Layer) of one of the body templates. (Once you have your working layer you can hit Ctrl+E to paste as a new "floating selection" which you can move around and manipulate at will.) Rotate the texture if needed so that the fur "runs the right way" on the arms, legs, or whatever.
Sorry for not providing a step-by-step on the above; not sure what's changed since PSP7, and it's good to mess around in a program like Kathmandu suggests, just to get an idea of what you're doing. A drawing tablet helps but is by no means necessary to start making good things.
For making your fur "tintable," what works best is to strip out the color information from your texture. I would do this in a copy of the layer, so if something goes wrong it's no big deal. Go to Colors --> Adjust HSL (Hue/Saturation/Lightness). Bring saturation all the way down. You may need to bump the lightness up a little -- use what you see in the preview as your guide. Colors --> Adjust Brightness/Contrast is also helpful as a next step. The goal would be to get the fur to a medium- to light-grey overall while still preserving the texture. Whiter areas will show through more "tint" and this is exactly what you want. It can usually be done without the HSL step, but if you try to "tint" a pre-colored texture the results can be unpredictable.
Lastly, use the game's preview option! The default, once you've selected your file to upload, is to "Display as Texture." That's in a pull-down menu; pull down to "Display as <insert body part here>". You'll see a little model of a head, torso, or whatever and you can click and drag to manipulate your view of it. If everything looks good, hit "Upload;" if not, "Cancel" and go back to PSP to tweak.
Hope this all helps. It is by no means the only way to create a fur texture but I think it would be easier than trying to draw one from scratch, at least for a first try. Good luck!