First, I think there a couple other things to consider besides your main question. You should probably have more than one screenplay before you seek representation. I'd recommend three. You are generally going to hear "what else ya got?" before people sign you, staff you, etc. People want to get a sense of your voice and your abilities, and they may not be able to do it with just one script. And don't necessarily think you need a broad portfolio of TV, feature, comedy, horror, drama, etc. It's kind of better, from what I've seen, to find what you do best and just keep doing that. Is your speciality The Female Superbad? Buddy comedies? Family movies with adult protagonists? Dark sci-fi stuff? Police procedurals? In a way it's frustrating that agents, execs, etc. will put you in a box and describe you with a buzzphrase, it will most definitely happen. If you can write a kickass drama and comedy, fine. Great writing is great writing, and it will get you noticed. Just don't force yourself to write something you're not passionate about.
Second, know that many beginning writers do not sell their first screenplays. It does happen, but what you should really focus on is getting an agent and getting yourself on people's radars. The script should prove you can write well. If it never sells or gets made (also two very different things), it can still be an effective sample to get you representation and rewrite assignments (which I will probably post on in the future since it's a huge part of the feature world that a lot of new writers don't seem to know about).
But now to your actual question: how do you meet people and get your writing career off the ground when you have no intentions of being a PA, assistant, etc.?
I once joked about my dream of writing by the pool while some attractive man paid for my cushy lifestyle. Of course, yours involves kids, which I'm sure makes you a bit busier. I honestly don't know how I'd be making all these connections without my job. But here are some thoughts: first, the internet. There are plenty of blogs, websites, tracking boards, etc. A friend recently started the LA TV writers (and friends) networking group, which will be having monthly meetings (there may be another one out there for feature writers).
I also recommend entering your scripts into contests; there are a ton for features, some of which are more highly regarded than others. The Nicholl is renowned as a big one. ABC/Disney also has a feature fellowship, and the Austin Film Festival has a screenplay competition. Google up some more - there are a ton. The idea with contests is to get noticed and get an agent; if your agent is well-connected enough, you shouldn't need to have a lot of your own industry contacts (though it never hurts, and writers who started as assistants probably already have a ton).
You can also send query letters to find agents and managers. I would say don't bother sending them to the Big Five, or even really the next five...but every so often you hear about that query letter that was so hilarious that a huge agent ended up signing the writer. It hardly EVER happens, though. You're better off trying smaller agencies and management companies; they will read them and occasionally request scripts that sound promising. Check the list of WGA signatory agencies for places to try. Beware of any "agency" not on the list, or an agency that charges you a reading or printing fee; it's a scam. Keep your letter brief, ideally just a few sentences. Don't go on about your degree or yourself; just give a a logline and try to make it sound as interesting as possible. I get query letters all the time, and if a letter was good enough, I'd send the writer a release and ask to read the script. Guess what? They never are. (They also usually come from Indiana or somewhere, and I know that if the writer was serious about their career, s/he'd move to LA.) People also have tendency ramble on and on about their lives, and often can't sum up their script in a few sentences. If you're passionate enough to write the damn thing and you can't even tell me what it's about, that's a problem. SELL IT to me. Because that's what an agent is going to have to do. Speaking of, here's some advice I heard my boss tell a manager the other day: "If it's not high concept, the writing has to be FANTASTIC." It makes perfect sense. You can sell a cool concept, even if the writing is absolute shit. They'll just hire someone else to rewrite it. But great characters and dialogue with a shitty concept? No one will buy it.
In Kathy's situation I'm also wondering if a little creativity in networking wouldn't hurt. Can you find out where female execs or wives of male execs get their manicures? Take Pilates? Enroll their kids in preschool? Do any of your friends know any of these people? It sounds a little crazy, but it might just work - and getting to know these people on a personal level might make them more likely to help you out.