Only works for about ten minutes, then if you shut it down it will work again in a bit? You mentioned that you had it checked and I'm guessing it had sufficient refrigerant not to be a concern to the professional who checked it.
What geographical region do you live in? Sometimes high temperatures and high humidity can make an A/C unit quit working when the evaporator core freezes up. Shutting it off, and allowing the ice to melt usually restores function to such a unit. BUT, it's not supposed to lower the core temperature below 34 degrees F.. This would lead me to suspect a low refrigerant charge or an improperly set expansion valve, which regulates the core temperature.
Converting over to R-134a is questionable at best, since R-134a is quite sensitive to volume in an A/C system, and many conversions are undercharged regularly.
As mentioned earlier, on another thread, GET A PROFESSIONAL to look at the system...don't screw around guessing this & that....
No good deed goes unpunished.