I have experienced similar problems when memory has been under perssure.

The OP said they have a 16GB box with 13GB allocated to SQL Server. This may be OK in the OP's environment, but typically allocating 12GB to SQL on a 16GB box is about the upper limit of what is safe.

Over allocating memory a small amount does not always cause SQL Server to run slowly, although large over allocation will affect SQL speed. However, lack of memory may affect other processes on the box that need to conmnect to SQL Server, such as SSIS and backup routines.

One troubleshooting tactic that is definitely worth doing in this situation is reducing the memory available to SQL by maybe 10%, and see if things become stable. If this does work, you can then increase the SQL memory in stages until things start breaking again, to prove that this is a memory starvation issue.