Many blinker systems rely on a certain amount of current through a set of bi-metallic contacts. When the light is on, it draws current and causes the contacts to heat up and expand at different rates (the reason for two different types of metals) and eventually break the contact, turning them off. As the contacts cool, the will eventually re-connect turning the lights back on. This happens repeatedly giving you a blinking effect.
If the new lights draw less or more current, it will effect the blink rate. If you have a burnt bulb (open circuit), the good bulb will stay on since there is not enough current from one bulb to heat the contact sufficiently. If the bulb drifts to a lower resistance from its design, they can blink fast as well since there is more current causing the contacts to heat faster. You can see this working on many Christmas tree lights that have a "blinker" bulb (usually clear with a red tip).
Judging by your symptoms, I'd say those bulbs draw a little more current than stock ones.
I have never looked, but do the stock driving lights come on at a lower brightness at all? If so, there may be a circuit to reduce the voltage to the lights to make them slightly dimmer. This is probably done via some type of voltage divider circuit, be it active with transistors, or passive through simple power resistors. If your new bulbs draw more current, less of the voltage and/or current will get to the lights making them look dimmer. This may seem confusing, but makes sense from an electronics perspective.
All this is based on the type of circuit I've described above. If the lights use an electronic oscillator to active a blinking relay, this does not apply as much. Either way, the new bulbs may require higher power (more amperage). they may fit physically, but I'd check the power/current requirements of each to see if they match. You should be ok with a 5-10% change in values, but you never know.
I really don't know how the system could be "rigged" to fix this, but the only solution I can think of is to re-design it and that is probably not an option. Maybe a Mazda tech on here can validate my post or offer more help?
Sorry for the long-winded reply, but I like to describe how things work so the person can visualize things easier.
