Only OTC medication I have found to help pain of Achilles tendonitis is aleve (naproxen). However, I have to take 600 mg. Is that safe on a regular basis?

Yes for short period. Of time, like 4-7 days or so. All nsaids are not recommended for anyone with kidney problems. Normal dose of aleve (naproxen) is 220 or 250 mg twice a day. For short periods of time up to 1000 mg maybe used. I would recommend pt for stretching and splint. If no improvement after 6 weeks, then maybe a MRI might be needed to evaluate the status of the achilles tendon.
Ice and . Physical therapy might enable you to reduce the dosage. Alleve is a lower dosage than naprosyn (naproxen) which is usually prescribed at 500mg twice daily. Speak to your doctor about your current dose.