Okay, so I ordered a repair kit from one website, and a new carburator from eBay. It took some time to arrive, but I finally got both. And yes, the main jet had fallen out of the original, because I though those parts was screwed in, which they normally are.
But it did not fix all problems. I now have a brand new carb from eBay, and the original that has been fixed using the repair kit. It runs fine and smooth, but ONLY with full choke on. This is regardless of the carb I use.