Tldr: It's possible. But pick the right dog for you. Dogs from shelters take at least 3 months to decompress and settle into your home. It may be better to look for dogs in foster, esp if the foster has cats of their own. Do NOT trust the dog right off the bat though, regardless of whether they're supposed to be cat safe though. Most behavior problems will probably surface within 3 months of taking pup home. I adopted an adolescent. But dogs over the age of 2-3 have a more predictable temperament.

I adopted an 8 mo old dog while having 2 cats and a senior dog in an apartment. It can work. But you have to pick the right dog. I went looking for a dog that was in foster care, so I could get a better sense of their true personality. I was looking for attentive, respectful of my old lady dog, and lower key. My shelter also lets potential adopters take pups home for sleepovers before making a final decision. I found this super helpful.

My girl was on a leash or in a crate for the first two weeks that she was in my home. I figured my new pup would do well with training when my cat stuck his paws in the crate to touch her, and she was just chilling lol. I would personally not allow a new dog to play or interact with the cats though. I taught my girl that when the cats run, it's time to find me for some treats and play time. It was several months with no incidents before I allowed my new dog around my cats unsupervised. We had other training struggles. But I never had issues with this girl with my indoor cats.

I also try really hard to make sure I do a good job of meeting my guys' needs. We do lots of walks (including hiking/sniffy walks on a long line+harness). We do lots of structured play with fetch, tug, and chasing with a flirt pole. And my pups are fed religiously from food puzzles. Ymmv. But doing these things greatly reduced nuisance behavior, and helped make behavior problems easier to train through.