Weeks of ice and eccentric calf stair stretches have eliminated Achilles pain but still always swollen, and tender after each run. Safe to run?

No. You have Achilles tendinitis and need a podiatry evaluation, support and anti inflammatory medication. If you continue to run on an inflamed Achilles tendon, there is a possibility of tendon rupture.
The question is. why? is it from exertional compartment syndrome? from venous insufficiency, degenerated achilles tendon. I think an MRI is a good place to start.