I found through experience that the caution against using first order elements in FEA is well-founded. Doing a convergence test on a simple rectangular plate fixed at both ends loaded with transverse and normal pressure forces showed that maximum displacement values converged with 10-100 times more nodes using first-order elements compared to using second-order elements. Wow.
However, I could not get maximum von Mises stress to converge due to the presence of sharp corners at the fixed ends, no matter the order of the elements. This apparently is a well known quirk of FEA for so-called 're-entrant corners'.
UPDATE: It turns out there was a dissonance between my intended physical model and the one I implemented in the *BOUNDARY card. All I had to do was add rotational constraints at the boundary. This was revealed to be the problem after I tried to fix the last two layer of nodes at each end.
Although the max stress is more tame now, the stress still does not converge completely. I notice that the max stress, after a certain amount of nodes, moves to the corner from the node adjacent to the corner. I consider the point where the max stress is still not on the corner to be the actual estimated max stress. This is also around the point the maximum displacement converges.
Thus I have found a satisfactory solution for my sharp corner convergence issue.
UPDATE 2: Perhaps a better convergence criterion would be strain energy. I have yet to test this.