NMEA can’t really be post processed because it only outputs a position. Most post processing is done looking at the pseudoranges to each satellite and correcting for errors in those.
Even with pseudorange information, multipath is a very difficult error source to eliminate. For you to meet your objective, you’ll need clear views of the sky (no trees, no buildings) with no nearby buildings to generate signal reflections. Even moving a meter can introduce large variations in multipath that can induce significant position errors.
Satellite constellation will exacerbate any errors you do have-- NMEA usually provides DOP numbers which will give you a rough estimate of how bad your geometry is. If you’ve got 8 SVs or so, you should be fine if you have a clear view of the sky.
WAAS capable receivers will have removed clock and orbit errors from each satellite and most of the ionospheric delay induced errors, depending on your location.
The remaining errors should vary relatively slowly in time and space. This means that if you capture one point, move a meter and capture another point, you should see a meter difference. If you capture one point and come back tomorrow to capture the next point a meter away, you probably won’t get the accuracy you desire. The longer you take and the further you move, the greater the error you’ll see.
Also, keep in mind that the error between any point pair might meet your needs, but if you chain together a number of points, the error between the first and last will depend on the time and distance between those extremes-- you can’t win by taking lots of closely space points hoping to bridge a large gap.
You can get a good idea of the errors you’ll encounter by putting the receiver in one place and recording data for 24 hours. Look at the position track over the period of time you’re interested in and see what kinds of errors you’ll see. This test will vary quite a bit over the course of years-- sunspot activity causes more trouble with the ionosphere and thus more ranging errors.
Also be aware that a lot of non-precision GPS sets have positioning filters built in that can lead to unexpected behavior. The vast majority of these are optimized for vehicle travel, not pedestrian. Some will average multiple measurements, some will integrate velocity data, some will stubbornly refuse to update your position until you’ve moved the receiver more than a few meters. Depending on what’s actually happening in there it may interfere with your “move 1 meter” test and may render my 24 data collection idea useless.
Probably more detail than you needed, but it’s not an easy question.