Lee seating die problems

Ah, like a handgun. Nope. Getting it started isn't the problem, the die making it worse is. I see what you're saying but the die has enough play in it for the toe of the bullet to be bulging off on one side hard. I even had one that was fairly straight, but it had egg-shaped the neck, just off to the side like a bayonet.
Get a new set of dies, it sounds like there is something wrong with them.

I have several Lee die sets and have not had a problem with any so far.
 
Frankford Arsenal universal bullet seater. I don't even use my RCBS Gold Medal Match seater die any more.
IIRC, @Riceguy put me onto the set.
Two stems, nine floating bullet sleeves (.308 is in the seater in the picture), micrometer adjustment head.
Available at Brownells, Midway, Powder Valley, etc. ~$80.

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You could try the seater stem from your 30-06 dies, and see what that does for you.

I've only ever had 1 issue with a Lee die (for the life of me I can't remember what it was).
Just contact Lee and they'll make it right.
Unlike RCBS.
My 284 Win die set was sizing the neck too much. As in if you stopped the bullet seating halfway in the neck, you could see a considerable bulge!
RCBS suggested that i bought the 6.5-284 dies, and to order the correct ones from them.
I really like the Forrester dies i got for the 284 Win!!!
:D

Anyways, give Lee a shout!
They have decent customer service!
Haven't had to deal with many die issues, but I had a problem with a Forster 'Bushing Bump Neck die'. When I called them I was ready for a fight (over the typical recent year's responses). Instead I talked to the most courteous rep I could have wished for, and the part was received quickly with no charge~!!
 
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