Have I broken or strained my ankle, it's swollen and bruised, can hobble around on it but it's so painful?

Be seen. I would definitely be seen just to make sure what you think us a sprained ankle is not actually a broken ankle. An exam and X-ray can usually give much more information about your injury.
Sprained ankle needs. to be seen and examined along with X-ray to see if any bone demage occurred and or ligament strain. See a foot ankle doctor as soon as you can especially since you have difficulty ambulating.