If I made a webstart, would I have to edit all the source code to work with it?
No. Webstart is very easy to integrate with a Java desktop application. I would definitely recommend using it, but you can wait if you want without having to worry about it.
While it can be easy to integrate with a Java app it can be a lot trickier to integrate with your build system or whatever release management process you may have in play.
I would highly recommend spending the time to learn it well and integrate it properly (instead of a hack-job) the very first time you need it (even for something as trivial as a demo to the java-gaming.org crowd). It'll save you a ton of time in the long run.