Your camera body is fine. A 10 year old camera body would be fine. Makes no difference at all. The lens can matter a little bit, in terms of producing certain looks you may wish to have. But even then, they're not all that critical either. You could do great newborn with an 18-55 kit lens. The key to newborn photography is lighting, atmosphere and processing. All the professional' stuff you see that people seem to be going for these days are completely unnatural looking (babies by themselves, naked, asleep, not cold, on some crazy fur blanket or dressed in some adult-esque constume, asleep, not cold, not fussing, in a basket in a field; if you really think about it, its ridiculous, but, it's what people want to see, thus it works); and these unnatural things are not popular and look like they do because of a specific sensor size or camera body, nor even the fancier fast fixed-focal length lenses (primes) with shallow depth of field and stuff. 90%+ or more is virtually all done with processing such as presets in lightroom, total environment swap outs, total sky swap outs, etc. The other 10% are doing it with lighting and props and some natural light and props and a lot of patience.
You have the tools. You just need to add more tools if you want to get on the bandwagon, by adding processing and props. Look at the trends, follow them, or create your own. But it will come down to the processing to make the difference. Everyone can take a baby, dress them however, put them in a basket or on a bench, in a field, or on some bear-skin or sheep-skin rug thing and snap a photo of them sleeping. The key is to do it when they're not fussing, not cranky, not cold, recently fed, not in a weird environment, and not being popped with high power strobes right on top of them (they startle them and they get hit with heat each time so its weird and can upset them), and simply get the rough idea of the portrait captured, then process it to be whatever you want it to be. Again, pull up the amazing angelic surreal dreamy newborn photos and take a look. 90% of it is processed to be that, if you saw the original, you'd be shocked.
Find out what you want to produce, professionally. Newborn photography is harder than people think, because newborns do not cooperate at all and you have to do it around their schedule, or you get fussy upset crying photos, poop every where, constantly peeing, and frustrated parents that are done with trying to do it for the 2nd or 3rd hour at their house, or in some studio. But you can do it if you work around them, find their routine, work into it, help the parents to understand that it doesn't happen on their schedule, and find out what they want the images to look like. Some want the social-media dreamy stuff (the unnatural looking stuff). Some just want something nicer than a cellphone snap. But a lot of it is processing on the dreamy side. For just good formals, you can get away with just some basic around the house stuff, good lighting, and some patience. But again, processing is mostly the big thing missing from what you're trying to do, just looking at your photos. Shallow depth of field and a fur rug or a tutu doesn't instantly create the dreamy stuff, as you well have found out. That's where processing takes the raw material and makes dreamy stuff from it.
Your equipment is fine and good for what you want to do.
You just need a new tool and skillset in processing and some more experience.
1) Gear is fine.
2) Controlled lighting is superior and cleaner.
3) Don't shoot auto. Control your camera. Don't just shoot auto and hope for something. This requires you really learn the exposure triangle.
4) Processing education. Pick a software suite (like Lightroom/Photoshop, or other) and start going through tutorials on how to do a lot of it. There are online classes. Subtle processing is much more difficult and effective, rather than very obvious processing that looks really hackish. But processing is the major difference here, along with experience. Not the camera/lens!