Let me see if I understand.
You have trimmed the case length back,so long cases are probably not the issue.
If you are trying the brass without seating a bullet,crimp,shoulder collapse, and seating depth are eliminated. Don't worry about those for now...maybe later!
Its a 6.5-06. Likely you formed it from something ,like 30-06?
And its a custom barrel/chamber job. You bought the rifle built.
You have a bunch of variables and unknowns.But those can be sorted out.
You might have to learn quite a bit.
IF you have a good riflesmith available(that's a big if) the most straight-line to your solutions would be a Cerro-Safe chamber cast to see what you actually have.
Next best would be if the seller provided a fired case with the rifle that is as fired.
But you may have neither.
Its going to take a process,one variable at a time,and take notes.
You likely are not using factory 6.5-06 brass. How do you get yours?
Are you forming 30-06 brass? If you neck down .044 thousandths,where does the metal go? The neck gets thicker.. Someone mentioned a "tight match chamber"
Maybe. Or your chamber might be the proper size,but with thick necks,your ammo may be oversize.One solution is neck turning.Sort of a PITA,but it can work.
I don't have the drawings handy.But look in load manuals and compare the .270 and 280 .There is some length difference on the .270,but see how the shoulder length dims compare. If you can use .280 brass,you are only necking down .020.Thats less than half the 30-06.I justed checked the .270.It has a longer neck,case length. Its a trim issue.The shoulder is in the same place (OK,1.948 vs 1.949) So you only have to neck down .013 if you use .270 brass. The neck won't thicken enough to cause a problem.
If you are sizing down 30-06,trimming and necking down .270 may solve your problem.( Something you can try,for a test.Your sizing die sizes the outside of the neck down far enough to be slightly undersize.The expander plug upsizes the ID of the neck for proper bullet fit.At that point,if neck thickness is too much,the neck OD will be oversize.So just to try to identify your problem,remove the decap/expander spindle from the die and size a few test pieces.Without the expander,the necks should be small enough.Let me know how that goes)
Next issue,headspace,sizing,shoulder setback.
Order a Wilson (or equivalent) bushing case length gauge. A 30-06 one or a .280 one should work if the don't have a 6.5-06. Now,somebody will disagree. Usually,they don't know how to use a bushing gauge,or the don't bother to maintain a known head clearance. I'll explain later.You decide,listen to them,or me. I don't care. I suggest telling them to ....excuse themselves. The distraction is confusing.
You trimmed and sized your brass.I assume you just bumped the shellholder with the die. Try it in both your new bushing gauge,and your rifle.
If it was tight neck from 30-06 brass,should feel better. Does the bolt close easily? Has the shoulder been setback enough?
Drop it in your gauge. Is it between the high-low limits on the gauge?.
Measure over the case and bushing gauge ,write the number down.
I don't care to write a book here,covering all the possibilities.
When you get this far,tell me the results. We'll take it from there.
A caution note: If,as I suspect,you have a problem with the necks thickening and being tight in the chamber,don't shoot any like that.It can be dangerous.The neck requires room to expand and release the bullet.If it can't,pressures go up.