Will I need to re-ring my pistons?

The issue is carbon on the intake valves and some white smoke. Valve seals. As to re-ringing it, I would run a compression test. If the numbers change drastically after putting a little oil in the cylinders I would probably make sure the bores were straight and round and dingle ball hone/re-ring. If no real change I would just address the original issue.
This is most likely the case. OP under what conditions does the white smoke come out of the exhaust pipes? If when decelerating or sitting still idling (with the engine fully warmed up), then it is the valves seals or some where in the intake. If under hard acceleration and not when idling, then most likely it is rings/pistons/bores. The seals 'looking' good' does not mean they are sealing well. And being 15 years old, they can easily already be hardening up. I had the valve seals start leaking on a 1970 Ford 302 in 1974 and at 70k miles... only 4 years from the factory.

So how much oil do you add and after how many miles? And do you have a PCV valve and what type is it? Also, do you know the piston type installed? PN or is it forged or stock or ???

Similar to the above, if the heads are still on, I'd run both a compression test and a leakdown test to get the best possible read on the ring and bore situation. The reason to do the leakdown is that if the rings are worn but are also oiling, then they can give good compression test readings without with adding oil to the cylinders. Then I'd pull a few springs, wiggle test the valves stems for excess looseness (not scientific!). If OK, then change all seals, and button back up and try again. Doing all this is not highly invasive and seals are cheap.