First, try html 4.01 strict:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
That is a badly-formed Transitional DOCTYPE, not Strict. The Strict DOCTYPE (the only one you should be using in this day and age, except for some very special cases), is:
Code:
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">
Alternatively, how can I get my page rewritten in one of the new doctypes? Is there a reasonably priced program that will convert it?
No. There are no programs that will convert it, because programs have no sense of semantics, which is the driving force behind the whole web standards movement 
I am a self-taught hand-coder, apparently from the 'olden days,' circa 1999-Y2K, when you could easily type out a basic webpage and it worked flawlessly with simple little self-explanatory html tags like font color and size, for fun a little scrolling marquee and maybe a few nested backgrounds.
Same (a couple of years earlier, actually). And boy am I glad we've moved on
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