All light meters have their own foibles. I've had a L308 for about ten years and previous to that I've used Gossens, Westons, Leicameters and several more that I don't recall. I've got the L308 wet in the rain and kicked it across several studio floors (black floor & black meter = bad combination) but it still comes back for more. I don't wish to sound rude, but it may be that you've set the meter wrongly or that your not pointing it at the tonal areas you're trying to measure accurately - you may, for instance, need to get closer to the area you're measuring.
It's also possible that your new meter is inaccurate, but I doubt it. More likely, you've established a 'zone system' that works with your old Weston but no longer holds with your new meter. For a zone system to work it requires a 'closed system' with consistent components. It may be that, in setting up your old system, your Weston was consistently inaccurate (compared to standard) and that you then - unknowingly - compensated for this in development... A zone system may work well within it's own confines but fall to pieces when you inject a new component ; it could be a new film or a new camera, or a new light meter or a combination of all three...
I suggest you discard the figures you've derived in your old system but, using the same methodology, start again - set up a new set of calibration tests and see where that takes you.