I beleive you are on the write track initially. Split the data and TLog files to seperate discs. The issue on having them both on the same drive is that they are comepeting. The TLog is being written to with every SQL Server operation so it is quite busy being written to during normal operations. Thus, this takes away from the resources needed to write to your data file (MDF). In a perfect world, you would even split your TempDB to another disc and other DB objects like indexes and such to their seperate discs/partitions.
In seeing your memory continually growing until exhaustion, I would take a look at what jobs, queries, and applications are hitting your DB during this event and determine where the issue is. Use Performance Monitor to pull some meterics as well as leveraging SQL Server Profiler. Also keep tabs on applications besides SQL Server that may be running on the same server. I would take a look at Task Manager for these items.
Hope this get you started in the right direction to resolve this issue.
Thanks
J. Kusch