Do I pillar bed or just bed the lug?

Savage 116 with Bell & Carlson synthetic stock.

I always pillar bed first. That stabilizes the gun in the stock and no more worries about take down screw torque. You can snug them or torque them, they are on a pillar so it won't matter. If that doesn't do it, I bed the action. In your case, I would check that the barrel is properly free floated. Take a twenty dollar bill, fold it lengthwise and slip it between the barrel and the stock. It should slide freely all the way down to the action. Do that and immediately send the twenty to me for confirmation.

What's that gonna cost you?

My limited knowledge...

Pillar bedding is achieved by installing metal pillars into the stock so when you torque/tighten down the screws holding the action into the stock the screw heads and bottom of the action don't squish/suck into the softer wood stock.

Glass bedding is don't to ensure that the action and stock have and "exact" fit so it goes back the same if you take it out.

Savages have a floating tang and from my research and understanding and are designed to stay that way.

I have a wood stocked savage short action I will be pillar and then glass bedding this summer.