Does it do it with new brass? Very common on belted brass to not get sized all the way to the belt and it starts sticking in the chamber. An M-16 extractor will not fix this if this is the cause.
All of my personal factory belted rifles do this eventually and I just toss the brass to get rid of the problem. Ruger, Sako,,Weatherby, Remington. They all do it if you run the brass hard enough or often enough. Someone makes a die that will fix this but I don't shoot my belted guns enough anymore to be much of an issue. I just toss the brass when I feel one getting sticky.
Stay away from Sako extractors on a 700. On a case failure they can become a projectile. The M-16 is pinned in and much safer.
Another problem putting a different extractor in a Remington 700 is the extractor ends up about 30 to 40 degrees too high and can cause the brass to hit the scope and fall back into the action. Here is what Remington does to help with the problem on their Lapua rifles where there isn't room for the regular extractor. They trim away part of the hook to get it to pull lower on the case. This problem is compounded by the ejector plunger being placed too high. It is factory placed to be straight across from the Remington extractor hook.
I would tend to just fix the factory extractor. It's most likely just bent a little losing is pressure. You can check this quick by removing the bolt and seeing if it snaps securely into the rim of a case. If it doesn't get much of a bite it can be rebent or simply replaced.