What is distortion due to?

Tough one. If it is more prominent in one channel than the other, then anti-skate needs to be adjusted. You did not say if you have a JMW-9 or Signature, but this is certainly the place to start. If all is well in anti-skate world, and I don't think that it is, you may need to increase tracking force to the top of the suggested range. Most MM cartridges put out between 3.5mv and 5.0mv, so with 2.5mv output from the Blacbird, you are certainly not overloading the phono stage.

I have been tracking at 2.0 gr, which is what is recommended.
It is a JMW 9, and the buzz is only on some records and is not just in one channel, so I doubt its anti skate.

I'll try a little more tracking force.

Could it be bad pressings?

First, it is probably a fairly serious problem, when the stylus loses contact with the groove wear is accelerated on both the cartridge and record. You say that it is not anti-skate, but have you put a twist in the wire from the tonearm to achieve anti-skate, and if so, how have you measured that the amount is correct? This is the trickiest arm to get anti-skate correct on, as it is so Mickey Mouse. I recently set-up a Dynavector XX-2 on a Super Scoutmaster and really had to monkey around until I got the anti-skate just right. Remember, if it is not correct, there is more force exerted on one wall of the groove than the other and side with less force will experience compromised tracking. My only other thought is are you measuring the tracking force in the plane of the record? Most scales require this, the exception is the, rather crude Shure balance, which is designed to work on top of the playing surface. And it could be bad pressings, but usually these make noise, but not buzzing, which would seem to indicate that the stylus is loseing groove contact. Are these the highly-modulated high-frequency passages that are problems?

If the noise occurs on both channels then antiskating and bad pressings are unlikely to be involved. Those problems tend to create one-channel noises.

Distorted musical peaks could indicate an overload, but this should never happen with a Blackbird feeding an A308's MM inputs unless the MF has gone bad somehow (pretty unlikely).

A buzz or static-like burst in synch with a musical peak usually indicates stylus mistracking. The solution includes keeping your records clean, keeping your stylus clean and increasing downforce.

Sumiko's recommended VTF for the Blackbird is not 2.0g. It's a range from 1.8 to 2.2. There is no reason to believe that "exactly" 2.0g is optimal for any particular Blackbird, especially one that's new. New cartridges often require more downforce until the suspension breaks in. Every cartridge is unique and many perform best in the top half of their recommended VTF range even after break-in.

VTF cannot be optimized by using a scale. It must be fine tuned by listening. Eliminating obvious mistracking is the first and most important step, since this kind of behavior will permanently damage every record on which it occurs.