Can I refer to myself as a Digital and Graphic Designer offering web and graphic design?

"Web and Graphic Designer" would do it for me.  No need to put "Digital," as that's assumed today.  But if you have skills with hand illustration, do reference that, as it's not as common a skill.

And of course, link a portfolio of your work.

Yes, you can say this.

"Web Developer and Graphic Design Artist" would be my suggestion.  I'm a "full stack" (and I hate that term) developer (xwlamp, etc. including multiple databases), digital graphics artist, software engineer and senior software developer for multiple operating systems.

"Digital and Graphic Design" is redundant, and kind of obfuscated.  If you actually can develop a web site, and are a digital artist, then what I initially said applies.

But always play with the wording until you feel it suits you.  Even if it's unconventional... sometimes just because it's unconventional.

So long as you're informed about the labels you choose and how they will be received by your prospective employers (or customers) then you will choose the right ones for your goals.  Learn and choose.

You could write a meaningful title for your skill and qualification. For instance, if someone has knowledge about photoshop illustrator designing, then the best title could be like this 'professional photoshop illustrator for graphics and logo design'

Take out the AND unless you also do graphic design using air brush or other analog art pieces. Or perhaps Digital Graphics and Programming.