HTTPS is the correct tool to use. The computational load of decrypting the packets is very low. Google changed to HTTPS by default for the whole of GMail earlier this year, and they report that the CPU load on their servers for SSL encryption/decryption is around 1%.
If you only encrypt part of the stream then you still have the problem of man-in-the-middle and replay attacks. SSL is the only way to prevent these. It doesn't really matter if the session ID is encrypted. If a man-in-the-middle can capture it, he can reuse it in it's encrypted form, and the server wouldn't know the difference.
Here's a blog post about Google's experience since the GMail switch to 100% SSL.