If you are looking for a word to describe a designer who works on both Look and Feel at the same time, without specializing in either one, you could call then a "User-Interface Designer", to refer to the fact that they are focused on making the experience of the user as easy as possible AND as appealing as possible.
Really, if a designer is working on a project at all, they should be concerned about both the look AND feel of it throughout the process (Unless, of course, it's a pure visual medium, but even then the manner in which it reaches the user could be refined).
If you're looking for a designer who manages and oversees the design process, such that they are in charge of both visual and interactive aspects of the design, you could refer to the item on which they are working (A Web Designer, or an App Designer), if it would apply to the situation. (If they're working on a particular part of a project, like a programmer who designs the first level of a game, you could refer to the part they are working on, IE, a Level Designer).
Really the only instance I can think of where a general Designer title wouldn't encompass the entire project would be in case there is an Audio Designer separate from the person designing the "Look and Feel" aspects, in which case you could refer to the first individual as the "Interactive Designer" to reflect the fact that they are designing the interactive portion of the project.
Hopefully one of these suggestions works for you!