Recently I was asked by a writer interested in film & TV if it was possible to have a career as a writer in those fields even if one lived outside the main production areas. Here is my response: I have not been involved in TV/movie writing for well over a decade. The little bit of video game writing I do is almost always the result of referrals by friends. I am concentrating most of my efforts on publishing now, specifically for the Christian YA market.
That being said, I will offer what insights I have.
First, while it isn't necessary to live where TV shows & movies are produced, you do have to go to where the show runners & producers are.
Fortunately, a lot of times they will come to you. When I was starting my publishing project about 10 years ago, I attended Book Expo America and what was then called the Christian Booksellers Association shows.
I did my research carefully, finding out (a) which publishing companies were publishing material similar to what I was going to do (b) which of those companies would be attending the shows (c) who the editors were at those publishing houses.
I then contacted them well in advance of the shows, briefly described my projects to them, and asked for a brief get-to-know-you meeting at the shows (there would be too many distractions to attempt to do any real business).
The objective was to become part of the landscape of the Christian publishing world. My first year attending the shows I was "Who?" but by my third year I was, "Oh, that guy" and by the fifth year I gained traction & began closing in on a deal (I'm skipping over a lot of give & take along the way since there are aspects of publishing & in particular Christian publishing that aren't applicable to film & TV).
Most Christian publishers in the US are in the South or Midwest. I didn't have to go out there to pitch, but as the deals became solidified I had to travel to meet w/them.
For your goal of writing for film & TV, I would suggest finding out where producers & show runners are going to be appearing, either as guests at film festivals or seminars, speakers at colleges & universities, convention panels, etc. Go there, introduce yourself to them, tell them what you've done, & ask for any insight they may have on breaking into the business.
Don't pitch a project on the spot. They will often be willing to help w/advice but will raise their shields if they think you're trying to pitch something.
The objective is to become a familiar part of their world, even if you're just on the peripheries. It takes a while, but gradually you'll start getting feedback that will steer you to people who might be willing to give you a chance to pitch to them.
(The alternate angle is to write incredibly good original feature film scripts, enlist the aid of a well placed agent -- again by seeking them out at seminars, etc. -- and having the agent find work for you. I am told that happens; it never happened to me & almost all my jobs & deals came through my own personal contacts, but it apparently does happen to some people.)
(And yet another angle of attack is to write in a non-film/TV field & develop a following based on your skills & abilities so that producers & show runners will seek you out. That works, too, but it makes the film/TV writing career ancillary to the primary gig.)
I would not recommend uprooting & moving to LA if you are married unless there is a support network of friends & relatives in the area & you can secure a steady side income until you start selling. Moving to LA w/o any writing gigs in place makes you a guppy in a very large pond; there will be literally tens of thousands of other guppies fighting for the chance to move up the food chain.
Instead of moving, first develop the connections mentioned above, try to build up a rep for yourself, and get producers & show runners to think of you as someone who can solve problems for them and/or make them money. That sounds cold & crass & mercenary but as Jimmy Durante famously observed, "Dem's da conditions wot prevails."
I assume you're already scouring writers' websites (include the WGA site) for more insight/info. If you aren't reading Mark Evanier's blog I suggest you do so: He often writes at length & much better than I do on this topic (& when he writes about something else he's always entertaining).
I'm sorry I don't have a name to pass along to you or the secret password/handshake. Breaking in can be a long and arduous process. If this is something you really want, if your life will be empty w/o it, go for it.