If you want to go down the non-parametric route you could always try the squared ranks test. The
For the unpaired case, the assumptions for this test (taken from here) are:
- Both samples are random samples from their respective populations.
- In addition to independence within each sample there is mutual independence between the two samples.
- The measurement scale is at least interval.
These lecture notes describe the unpaired case in detail.
For the paired case you will have to change this procedure slightly. Midway down this page should getgive you startedan idea of where to start.