I wonder why you're so desperate to avoid "drive."
And just in case you're interested, this way of saying it (apparently common in Australian English) would be unusual in American English. An American would likely say, "I'm going home," and that's all; we wouldn't generally bother, nor would we usually see the need, to mention how we're getting there, unless there were some special importance to the mode of transportation, as in this example: "I'm going home, and I'm not taking the subway. I've got my own car now!"
There are also creative ways of "handling this sentence," as you ask us to do. You can completely restructure it if you like, and here are some examples:
"I'll just hop in my car and head home."
"I need to go home. Where's my car?"
"I'll take my car and make my way home."
"My car and I are going to team up to get me home."
You can get into some fairly strange constructions, as you can see, which can be used to say the same thing, if it works for the story you're writing.