In these two measurements are you using the same power supply? You say you use a pot and it was good but using the LM35 you get erroneous voltages.

I wonder if when troubleshooting you used different power supplies and the DC voltage may be from an analog supply in the first case and a switcher the second. If so the switcher may need an electrolytic on the supply to the LM35.

In the last part of your post, how and where are you measuring the voltage? Could your meter/probe be loading the output of LM35? It would be interesting if you'd show some sketches of how you measured those voltages and what's connected to the LM35.

Just some thoughts.