Hi. Your post read like a blast from the past. I started out in a Ruby on Rails shop with an internship making $10 an hour. My laptop was an old HP, with fans that whirred (for some reason HP can't make/procure fans to save their lives). Everyone else had sleek new equipment. I built a couple great features, squashed tons of bugs, and learned a lot along the way. Near the end, I talked with the founders about raises a couple times, but it's clear what they needed was a way to save money on development for as long as possible. And what I needed was the experience of working with a real software team.
We all got what we were looking for. It's just a question of when you don't need that basic training anymore. Sounds like that time came for you around the same time it did for me. I can tell you - at that point, the only way up is out. You are much, much more employable now. Don't "beg for work" ... present yourself well and ask for market-rate. You said that's $75,000 in your area? Better yet, say you're looking for a range of $80-90k and see what they say.
I've got a question for you, and maybe an idea. Did you sign a non-compete agreement beyond just your term of employment?
Even if you did, it might not be enforceable in your state. Go to the billion-dollar rival and tell them how you've got almost a year of domain expertise under your belt working on a similar product. You're looking for that next step - full-time Software Engineer, and a chance to build on what you already know. You're able to make clear comparisons between the two offerings, and I bet you have some ideas about how theirs could be the best of both worlds. Oh, and when they ask you...
"Why did you leave your previous job?"
Don't panic. 18 story points on your last week, remember? You're a sharp web developer who rapidly rose in value - far faster than your perceived value to a CEO who forever saw you as "the cheap intern." Since you couldn't reach an agreement that met your salary requirements, you parted ways. Now you're able to devote more of your time to the search. That's all you need to say - so practice saying it with confidence.
Email real people at the target company (email is the new cover letter) or start asking questions on Twitter - don't just drop your resume in an upload. Start a conversation. Get your Team Lead and as many coworkers as possible to jump in with recommendations, when they ask for references.
My next offer after the internship was a real one, at market salary in Silicon Valley. There's light at the end of the first-job tunnel. It's tough now, but please believe me when I say it does get better.