Encoding video is a very CPU intensive process (not memory intensive, so changing memory will not help). Your computer is running relatively slow processor (2.0Ghz Celeron), so its not surprising that it can only handle one stream at a time, especially if you're using HD video.
To fix the problem you could look into external hardware to encode your video, or into a more powerful computer. Another option would be to decrease the quality of the video (maybe drop down to just 480p instead of 720p).
Update:
After thinking about your problem for a sec, what you really want to do is have a single application encode the video, and then stream the encoded video to two different streaming services. This should be possible if your machine is able to stream to one without problems. I'm not familiar with how streaming websites accept streaming data, so I can't help you with implementing it, but something like VLC which can output to multiple sources might do the trick.