Before you do anything else, make sure your sizing die is set up correctly. When a lubricated case is pushed up all the way into the die by your press, you should not be able to see any crack of light between the shell holder and the bottom of the die. See
this Lee video for an example of how they do it with their die. Except for the Lee finger-tightening locking nut, it is the same as for all standard dies.
Below is my illustration of what Wogpotter is talking about. Note that the two bullets are the same distance from jamming into the throat (same distance off lands), so both are seated correctly, but their COL's are quite different due to the bullet shapes being different.
The usual poor man's diagnostic tool for feed issues is the Magic Marker. Make up a dummy round (no powder or primer, but same bullet seated as you've been doing it). Color its bullet and case with the marker, and after it dries, chamber and extract it and look for rifling marks on the bullet and rub marks on the case. If the bullet has rifling marks, it needs to be seated deeper.
To get a better idea of what's happening, you can use your calipers with spacers as gauge tools and check your case against a new case or, if the bullet fits in the spacer, a commercially loaded round, as shown. Also sub in a spacer with an oversize hole for 1/4" bolts that is just under the size needed to let the .308" bullet fall through, and apply that to a loaded commercial round and then to your own loaded round (the dummy). You want this measurement to match in the beginning (when you get more advanced you can fiddle with this to improve accuracy, but don't mess with it at first).
Note that the measurement is made after closing the caliper on the spacer and pressing the zero button. The resulting measurement is slightly short for a .308 because the hole through the spacer is slightly larger than the SAAMI standard 0.400" datum diameter for this cartridge's case shoulder. However, you can still get a good comparison between cases this way. In particular, measure your cases before and after resizing to make sure they get at least 0.002" inches shorter after resizing.
If the chamber is tight you may, indeed, find a small base die more convenient. However, before you go there, take a case you've resized and decapped normally. Remove the decapping pin and expander from your sizing die, lube the case again and slip a .002" thick automotive feeler gauge under the case head in the shell holder and size it again. That will make it at least .002" shorter (about what a small base die does, though it usually comes out a little shorter, still, just because it is making a second trip into the die overcomes some of the prior elastic rebound). Put the decapper/expander back in the die and lube the inside of the neck (graphite powder is good enough, but you can use wet case lube, though I like to alcohol that out with a Q-tip or tumble in plain corncob afterward to remove it).
Recolor the case with Magic Marker and try it again.