I wouldn’t recommend shapekeys because you’ll meet some problems as soon as you change your topology, or combine several facial expression, etc… it’s ok for very simple objects or for some body deformation but for facial expressions you’d better use bones. About your topology problem and mouth shut, perhaps share your file? https://pasteall.org/blend/
i’ll share the last three so you can see what i’ve been doing.
the last one has the mouth shapekey, but not the eyelids since they weren’t working.
not bad, add a bit of topology (more edge loops), keep a good edge flow (see how it rotates around the mouth), avoid triangles, and you will have a nice round shape, I’ve reworked a bit, it’s not finished of course (also use a Mirror so that you only have to work on one side):
so should i do all that before asking about the next step?
what is your next step? yes I guess it’s better to have a good topology before continuing with your project?
my next steps are what i showed in my blend files