It's not a matter of having a "flat map" which you then "put in a projection". That's a common misunderstanding.

All maps are flat, and they are representing a curved surface; this means you have to distort the curved surface by stretching, squashing, or tearing. The particular way you distort it, is a projection. So all maps are flat, all maps have a projection, and all maps have distortion. (Unless the world is flat)

The tricky part is that in order for everything to work out, you have to draw whatever distortion is appropriate for whichever projection you are starting from. If you don't, it means you've effectively given the land itself the opposite distortion of that caused by the projection. You appear to be using Plate Carree which stretches everything out east-west as you approach the poles. You haven't drawn the features that way though, which means you've drawn land that is 'pinched', as you approach the poles, and the projection then stretches it out so it doesn't look pinched. When you switch to another projection, you'll get that backward distortion combined with the distortion of the new projection.

Also, you shouldn't symbolize your map until it's in the final projection otherwise the symbols will get distorted. If you want to make this Winkel Tripel, you should stick to just the raw geometry of the features (solid colours, lines, etc) until you project it into Winkel Tripel, and only then do you try to make it pretty, otherwise you're just wasting effort you'll have to redo later.

God job picking up on what "small scale" means though, a lot of people new to cartography/geography get that backward.