My question is will I be able to get a mortgage with good rates with the scores I have now?

I am 23 and my credit scores are: 582 Trans Union, 552 Experian, and 545 Equifax.

I have a number of medical bills that went to collection while I was in colllege. I have been paying those medical bills back a little at a time (I have a payment plan with the collection agency). And have two 30 days late (almost a year ago) on my car loan, and a 60 days late on a student loan.
I recently applied for a new secured credit card (which I hope to receive any day) and have been listed as an authorized user on my cousin’s credit card. My cousin has had her credit card for 2 yeas and has never been late.

I am currently working with a law firm that has actually gotten 6 items deleted from the three reports. So, my scores are a little better than they were.

I just feel that it is not right to be punished for getting sick (I had cancer) and not having insurance.
My current scores are 559 TransUnion, 618 Experian, and 557 Equifax.

The reason the Experian score is so much higher is that a new revolving account (my cousin’s credit card) has reflected on that report. I am expecting the credit card to reflect on the other reports soon.

If the other scores jump anything like the Experian score, I expect the others to be around 610.