You will still have issues if you are full length sizing. The first thing I notice if I don't anneal when I should is inconsistency in my case sizing. My accuracy also degrades. If I don't take care of it when I see that, I will start having some cases fail, as in split necks. Not sure of the exact reason it happens but the necks definitely will split sooner in my 7RM than the rest of my rifles.
This. The brass will work harden (stiffen up), then become brittle and lose it's elasticity. You need the brass to have some spring back to hold the bullet consistently. Some calibers/bullet diameters work harden much faster than others. 6.5mm or smaller will require annealing more than say a 308 Win will.
Once the brass begins to work harden (often after a firing or two), the neck tension will vary from case to case and not be consistent. And your extreme spreads will certainly increase. If you anneal them, the brass elasticity will return and be consistent. If you are loading for a precision rifle that shoots very tight, you certainly notice an issue. If it's a hunting rifle that shoots decent, you may or may not notice it.