We should shoot the waste into to sun.
(Earth orbits the sun at 30 km/s [1]. That means you need -30 km/s of ∆v to kill the energy we're born with. The escape velocity for our solar system from Earth's orbit, meanwhile, is about 42 km/s [2]. So you just need 12 km/s of ∆v to skip town.)
[1] https://en.wikipedia.org/wiki/Earth%27s_orbit
[2] sqrt((2 * 6.7E-11 * 2E30) / (150 * 10^6)) given G~6.7E11 [a], mass of the sun is about 2E30 kg [b] and the Earth orbiting the Sun from about 150 million km [b]; for escape velocity [c]
[a] https://en.wikipedia.org/wiki/Gravitational_constant
Once you're out in the Oort cloud on the way out, you can't exactly deploy a solar sail to get another push (unless you make it impractically large).
It's much cheaper to fling things out of the Solar System than into the Sun.
Thrust available to a spacecraft using a solar sail is a function of distance to the sun. The sail is a fixed cost, but it may be presumed to degrade over time. It has no propellant that costs additional money to launch.
If you go in, the ever decreasing efficiency of the sail is offset by the greater available energy from the solar wind, and you may still be able to complete the mission (more slowly) with a damaged sail. Your available thrust increases with every kilometer closer to the sun.
If you go out, the decreasing efficiency of the sail compounds with the lesser total energy available to the sail, and if the sail is damaged, you may never reach escape velocity at all. You will never have more thrust than the instant the sail deploys.