Model converge at high loss (good fit bad performance?)


a: My model converges and looks like a good fit (training and validation loss follow each other closely)
The problem is, they stop (go horizontal) at a value (0.15 something) that’s higher than I would like (0.05!)
Since the loss converged, I can’t reduce it by training more (e.g. if I increase patience of early stopping, that just lets the model overfit)
b: my model is a good fit only in cross-validation. But when I go to testing, the model overfits (5 to 10 times the loss)
I’m confused about this, since the point of cross-validation is to make sure that the model generalizes well. In this case, cross-validation tells me the model generalizes well, but the testing tells me the model does not generalize well.

For problems ‘a’ and ‘b’ above, what should I try?

Many Thanks!