Interesting... I also frequently lost the bead on my 700x23c road bike tire back when I started riding. It actually turned out that I was losing the bead because I was UNDERINFLATING the tires.
Back then, I only had an el-cheapo Schraeder-valve tire pump made by Bell. It was leaky and it had no gauge, so I was simply pumping the tires up until I can feel it get "firm" by squeeze-feeling it with my fingers. Turns out that was not enough-- I would hit a bump, feel my wheel wobble, come to a stop, and see part of the bead dislodged from the rim. I then had to deflate the tire to get the bead to seat again, and reinflate with my portable pump.
All that time I thought getting the tire pressure to "firm" was good enough. NOT EVEN CLOSE. "Firm" was only 70 PSI when the 700x23c tires had a MINIMUM recommended pressure of 110PSI (printed on the sidewall)!
I bought a new Specialized floor pump with a pressure gauge, and now I inflate my tires to 120PSI before every ride (5 PSI below the 700x23c tires' maximum rated pressure), and I've never lost the bead since then.
In your shoes, I'd get some thick tubes and always inflate to the recommended pressure range using a pump with gauge.