Lots of good info so far, but after reading everything, I say your best bet is simply to bite the bullet and go out and get yourself a router. You can get a good Craftsman router for under $75 new or do a little searching and find a used one for less. I've had my Craftsman router for years and have used the heck out it. I'm sure it has paid for itself many times over.
Of course you make the joints (T&G) with hand planes, but you'll spend more on hand planes than on a router.

As far as what joint to use for your paddle. T&G is probably your best bet, dowels or biscuts probably wont allow you to shape the paddle exactly the way you'd like because of the thickness. And neither joint can be made successfully without other (more expensive) equipment, either a drill press or a biscut jointer.

Of course you can make a good biscut joint with a router, which leads me back to my orginial point. GOOD LUCK!

Let us know what the final solution is.

: Thanks for the encouragement, Chip. It's very much appreciated.

: Do you suppose a couple of dowels or biscuits would work and be just as
: strong? I hadn't thought of something like that. Don't know how to cut
: slots for biscuits, but I can certainly drill holes for dowels. I'm sure
: lining the holes up straight won't be easy either, but it's got to be
: easier than what I was trying to do.

: I live in Cary, NC which butts up against Raleigh.

: Patsy