The correct sentence would be just "I" saw the sun rise as I drove home that morning. I've is a shortened term to I have. I've got a blue hoody on. Hope this helps =)

I agree with what's been answered. One additional use of this phrase: "I've got to go home" means "I have to go home". That is, I need to go home. It is required that i go home.

You could say "I got to see the sun rise as I drove home that morning"; it is vernacular, informal. It doesn't mean precisely the same thing as "I saw the sun rise as I drove home." When you use, "I got to" do something, the implication is that it's a privilege. "I got to go backstage and meet the band." "I got to sing my favorite song at karaoke last night". Also, no pluperfect when the actions occur at same time (driving, seeing sunrise).