Look at Casio digital cameras. I have one for working on golf swings that does up to 1,000 frames per second and slower. At 1,000fps, obviously you need a lot of light, but it does really well at 200 fps or so in less light. It's not a "camcorder" but rather a still camera. They designed software that overcomes shutter lag by buffering when the shutter release is partially depressed, and in the process ended up with a great slow-mo camera.
I think it was somewhere between 200 and 300 dollars. Mine is several years, and generations old, but it does remarkably well. It's a point and shoot without interchangeable lenses, but if you don't stand too far away, it will work fine. The wide angle to telephoto lens does better than you probably think it could. We also have a DSLR, and this little thing takes just as good pictures in ordinary conditions.
If I'm remembering correctly, mine is an FH-100, which is now discontinued, but I'm sure they have similar ones now.
If you do a search on golf forums for high speed cameras you can probably read about the latest and greatest. I haven't bothered to keep up with them since I bought this one. It also takes really decent pictures at 10mp for a pocket camera.
I'm also looking for a video camera where you can take get good quality stills from the video. I want a device that can also function as a regular camera but I'm primarily interested in the video capability. Is there anyone with experience with a video camera they'd like to recommend?
You'd want something with high-FPS burst shooting at high-resolution, which isn't something consumer cameras are capable of. The current crop of consumer gear can only shoot high-FPS at sub-YouTube resolution because they do not have the processing power or memory to do it at high resolutions. DSLRs are starting to get there but they are well out of your budget.