Do brush her teeth. One of mine has gorgeous teeth. The other not. I have to not only brush daily but give her a dental twice a year.
I've had a couple of dogs that ended up toothless. While they still ate just about everything, they had the pain of bad teeth, pain of extraction and struggled with the hard treats.
Too many baths will dry her skin more. Try adding fish oil to her food. Or eggs. Since I have chickens they get eggs regularly.
Yearly checkups are plenty unless she has prob.
Also they are usually easy to do nails. Search for dremel in this topic or in EVERYTHING to learn how to dremel their nails.
Does she have fleas? I'd treat only if needed. But if you have heartworm disease in the UK definitely give her monthly meds for that.
Also, there are great books for caring for greys. One is Care of the Racing Greyhound, another is Greyhounds for Dummies. (I hope I got the titles right)
And congrats on joining the greyhound cult!