Disclaimer: I'm extremely green (read: n00b) when it comes to simulation, however I am working on a fairly complex analysis myself right now which has me studying a lot on adaptive meshing to better understand my own problems. My thoughts offered are simply my interpretation of what I've read on the subject and are not based on much experience.
One thing I'm a little confused on in your original statement was that you said that solving with H-Adaptive ON always failed.
You then said you finally found a solution that solved.
So, was this solution finally found with H-Adaptive meshing turned ON, or did you play with mesh controls manually until you got something that worked?
I'm just wondering, if you found it by manually playing with mesh controls - have you tried running another study with adaptive meshing since you got this solution?
Couple of questions on your adaptive mesh settings:
Have you tried bumping up the maximum loops for H-Adaptive?
What about decreasing the accuracy tolerance (e.g. to 90-95%)?
Have you played with the bias setting at all? Looking at the stress distribution in your screenshots, my first thought is that a fairly heavy bias toward localized accuracy would probably be acceptable.
Lastly, if your concern is specifically with the nodal vs elemental discrepancies, have you tried solving with the P-Adaptive meshing option?
I would think p-adaptive might help with this particular concern, as you may need better optimization of element shapes to get the nodal-elemental convergence desired.
Best of luck!!