When you have the image you want, you can click on File > Export > PNG Optimizer. Or you can have PNG Optimizer on your toolbar.
When the optimizer opens, click on transparency tab, and choose either Single color tranparency-which will leave the edges a little rough, or Alpha channel transparency, which will leave a nicer image.
These both have transparent backgrounds, though they may show as white. This works for Paint Shop Pro 7 through 9, I don't know about earlier ones.
Note: PNG alpha transparency is not supported in Internet Explorer 6 and earlier.