I would say there is a very slight difference in meaning. As was already mentioned mood is usually mentioned as a positive or negative emotion. He was in a terrible mood after work. She was in a good mood all day long. But morale is a word that describes someone’s overall sense of well-being. Your morale would be low if you were down in the dumps (depressed). A team’s morale could be low if they were losing a game. An employee’s morale could be optimistic if he thought he might get a raise.

"keep her mood high" should be "keep her in a good mood" mood is usually used with good or bad. She's in a good mood/she's in a bad mood. The first is correct but I will list a few that would be more likely heard: We do our best to keep her spirit up. (if she was sick or sad) We do our best to keep her morale up. We do our best to keep her in a good mood. Hope these are helpful. One can also be in a foul mood but I think our cousins across the pond use "foul" and we in the US use "bad"