They may already have a certain amount of infrastructure, like ballot access and fund-raising networks. You could probably get into officially-sanctioned debates and town-hall meetings. You wouldn't have to reinvent the wheel.

Theoretically, Libertarians are socially liberal, fiscally conservative.

And they are often looking for respectable candidates. As a practicing attorney, they'd probably give you some serious consideration. If you are actually serious about running for office, this might be a good place to start..

It is impossible for a political party to be socially liberal but fiscally conservative. Being socially liberal in the U.S. means that you support the government cleaning up the messes people make of their own lives. That requires a huge amount of government spending. there is nothing fiscally conservatives about the Democrats (See California).

I'd love to form a third party- socially liberal and fiscally conservative.

Why? that sounds just like the Democratic Party!