The only solutions I've found are expensive, and involved a pricey piece of equipment called a beer gun. Is there a way to carbonate my beer before bottling so I can ensure a good carbonation?
Well you carbonate the beer in the keg the same way as if you were going to serve from the keg. There is no carbonation procedure on the way into the bottle.
To get carbonated beer into the bottles however, the cheapest way to do it is to jam some 3/8ths inch tubing onto the end of your picnic tap. Using about an 12 inch piece of tubing you can put the tubing all the way to the bottom of the bottle and fill slowly. If you use cold bottles you'll lose less carbonation.
I tend to turn off the gas and turn my regulator down to near zero pressure. Bleed the keg completely. Put the tubing/picnic tap device into the bottle and depress the tap. Then slowly open the regulator until there is just enough pressure to see the beer start flowing. Fill 3-4 bottles at a time and cap them off.
Thats the cheap way I do it. I tend to carbonate the beer a little more than I would normally serve it at. Just by a few more pounds on the regulator during the carbonating process. That way I lose the extra upon fill.
Never had a judge complain of oxidation or under carbonated beer, and I am only into it for the cost of 12 inches of tubing.
This is a good design for a counter pressure filling device, it used the same principals as a long tube bottling filler uses in a brewery. I think these designs can be refined or modified but are a good starting point. http://www.ipass.net/mpdixon/Homebrew/CounterPressureFiller.htm
There are two parts - carbonation, and getting it in the bottles.
For carbonation, there are various methods, but I use the set-it-and-forget-it method. Beer goes in keg, keg goes in fridge, CO2 gas gets put on keg. Just set the pressure to the amount of CO2 you want in solution - "volumes" of CO2 - based on the style of beer and a handy temperature / pressure chart like this one. Then you wait until it's carbed to your preference. I usually wait 2 weeks for full carbonation. Here's another link with a style guide.
Now to get the beer in the bottles, I prefer a variation of the Biermuncher Bottle Filler. It's just a picnic tap, hose, rubber stopper and racking cane. You basically jam the racking cane in the picnic tap, then put a stopper on the racking cane. The stopper allows you to keep backpressure on the beer as you're filling.
I do it right off my taps using an adapter for my Perlick faucets. I've made two videos on the process I use. It's convenient and cheap. I can bottle right at serving pressure (12psi), and I don't have to move kegs or change any fittings in the fridge, which is a big plus - the keg I want is invariably all the way at the back.