Hi, English isn’t my mother tongue so I was asking myself that question since I first encounted a w/… Back then I was like: “What tf does ‘w slash’ stand for?” And when I found out I was like “How, why, and is it any intuitive?” But I never dared to ask that until now
I don’t think writing “the Dr told me X” is normal because it’s in the middle of the sentence. Especially if it’s capitalized because that signifies that it’s a title, but you’re not using the title (unless we’re talking about German which we’re not).
You should just type out “the doctor said X.”
In formal communication I would type it out, but not, for instance, texting my kids with updates on the cat or their dad or whatevs