I am trying to add a few more 240V circuits for a welder and plasma cutter in the shop. The 200 amp panel is already full (4 30amp, A LOT of 20 amp, and a 100 amp service piggybacked). I have a new 200 amp service set at a barn about 50 ft. from the shop. Can I run another 100 amp service into the shop off the meter base/breaker panel? If so what is the best method and how deep does it need to be buried?

No. You cannot have two services brought to one building.

No, but you can feed a sub-panel from your service. Re-route at least one of your existing 240 volt branch circuits to the sub-panel to free up breaker space at the main panel.