Buying a capture card would only benefit you if you had a 2nd pc to stream to. So you would stream pc 1 to pc 2 and pc 2 would stream to the internet. This is only good if you are running ultra high settings and want to stream at 1080p or higher with all settings at max. Because you'd have an entire PC dedicated to just encoding and streaming.
The pixelation could also be a result of the settings of the program you are using to Stream. I stream with OBS and I have to tweak a bunch of settings in order to reduce/eliminate pixelation. If your PC specs are similar to mine and you are using OBS I can send you my settings.
So unfortunately you're stuck in the same problem I am. My max upload caps at like 1.25Mb/s. So basically you have to downscale everything. So big things to mess with are, bitrate, resolution and FPS.
Bitrate will need to be 1000 or less(depending on other traffic coming from your house, you probably won't get more then 600-800 available upload constantly for you.)
You can choose screen size 1920x1080, but you need to downscale to 720 or 480p. You will not have enough bandwidth for 1080 and probably same for 720p. When I stream, I can only ever stream using 480p downscale.
FPS- 25 or 30. Make sure you do not go more then this.