First, I would find a way to afford a seat for her on the plane. It just is not worth the risk to your child safety to have her as a lap baby. Not just in crashes or crash landings, but even bad turbulance, you could loose your grip on her and she could be injured...or injure someone else, as she would become a projectile.
The link below is hard to watch, but worth it.
http://www.youtube.com/watch?v=rC4Q3ofz7SI
Another question is, do your really need the stroller? Are you visiting family or just touring alone? Is there someone you could borrow a stroller from when you get there? Is there someone in California that could go buy a really cheap umbrella stroller for you to use while there, then donate it when you leave?
The way I have flown (alone) with my son, has been to strap his Marathon to a small carry-on luggage using the Traveling Toddler. The strap is about $15 on Amazon. It securly straps the carseat to the luggage, and you can use that as a stroller at the airport. My son LOVES riding around that way. Also, I would bring a backpack as a carry on. I was able to travel alone for a week with just a backpack and the small carryon...but I did it because there was someone on the other end buying diapers for me. If you needed to check another bag, bring a wheeled one. it would be easy enough to wheel them both to the check in area.
As far as boarding the plane, I unhooke the carseat from the wheeled carry on, at the door to the airplane and have the carry on stored at the front on the plane for me. Then I carry the carseat onto the plane. Normally I let my son run ahead of me down the aisle while I carry the carseat, but there were sometimes when de-boarding that he was asleep in the carseat, and I was able to unistall the carseat and carry it whole thing with him in it off the plane, hook it to the carry on at the front of the plane and get to our next gate before he ever woke up.
The reason I unhook at the front of the plane is because the Marathon does not fit down the aisle and has to be carried above the armrests of of the plane seats. I believe the roundabout may fit down the aisle, but I might reccomend the unhooking at the front of the plane anyway, just to avoide a traffic jam on the plane w/ other passengers.
Also, if you weren't planning to already, keep your baby rearfacing on the plane. Not only is it safer, but she will sleep better and be easier to be entertained.