Lucy is shrewd. She behaves differently with different people. She knows on whom her flattery is effective, and on whom it isn't. When there is something for her to gain from a situation, she tells people what they want to hear.

I can just imagine how she got Edward to propose. :-p

Is it that LM and Lucy understand their faults and both could improve themselves if they wished?

Yes, but they do not wish it. Instead of improving their characters, they simply modify their behavior when they are around people who won't approve.

Does it mean they secretly desire the good opinion of the Miss Dashwoods?