Is drawing a necessary skill for becoming a successful civil engineer? Can an engineer be successful without strong drawing skills?

If you are talking about the ability to literally draw things with a pencil - not a necessary skill. Helpful, sure. But not necessary.

if you are talking about drafting (“drawing” in a computer aided drafting program), then yes, that really is a necessary skill. You can be great at calculations, but nearly every project needs to have clear contract plans/drawings. You need to be able to create them as a young engineer, and be able to review/read them as you progress in your career.