I think it's very likely that lighting is a significant contributor to your problem. Your camera phone has a small sensor which needs a lot of light to get top-notch results, and and indoor lighting provides a lot less light than most people think. So, to compensate, it is probably:
- holding the shutter open for a relatively long time, which introduces motion blur, both from your subject (or you, if these are self portraits) moving, and from not holding the camera perfectly still
- amplifying the signal, which introduces visible noise, which is probably then smeared out by built-in post-processing
Using your existing DSLR is a good move, because this gives you more control. I'd suggest following some of the basic questions here, like What is the relationship between ISO, aperture, and shutter speed?, to get started, and maybe buy a couple of books if you learn that way. I recommend anything by Michael Freeman — Michael Freeman's Photo School Fundamentals: Exposure, Light & Lighting, Composition is probably a great start for what you want to do.
Switching to your DSLR won't be magic, but will significantly expand your options. You will have better control over aperture and shutter speed — and even if you use it in auto mode, you will be able to better see what decisions it makes to get the exposure right. You might also want to buy a second lens suited to your use — I'm assuming you just have the included "kit" lens.
But first, you should buy a flash. And you should definitely do this before you spend any money on a different camera. The Canon you are looking at is only slightly newer than your Nikon, and has no fundamental advantages at the level we're looking at here. (If you just like it better, that's fine, of course.) And, this goes for buying a newer fancier camera too — sure, the specs will be better, but you don't need it to solve your problem. What you need is to take control of your lighting.
You can start with just one flash — I recommend something manual that you can radio-control from the camera body. This gives you the control you need, is very cost-effective compared to high-end automatic flash, and will be system-independent if you do decide to switch from Nikon to Canon (or to Pentax or Fujifilm or Olympus or Panasonic or etc.).
There is a fantastic online tutorial for basic lighting with a single flash in this way called Strobist 101. Go through that. That guide has some basic recommendations for gear, and see also What's the best bang for your buck to improve low light portrait shots: Lens, Flash or Body?