Can you post an example progran that does function correctly in X-Carve? My guess is that you either have mechanical issues and perhaps have been lucky when using Easel, or you need to get a post processor that X-Carve understands. Think about it logically:
1) Easel is meant to work with X-Carve, and has been optimized to produce toolpaths that X-Carve understands.
2) Fusion produces code that works fine on machines with 100 times the precision of X-Carve, so it is 100% impossible for Fusion to be generating toolpaths off by .1-.07 inches.
3) Fusion is using the same math to calculate toolpaths for every application, which occurs prior to post processing.
4) There is no optimized post processor available (that I can tell) for X-Carve.
If those facts are true (I'm positive about all but number 4), then the problem is one of three things.
1) User error when programming in Fusion (not likely, but a posibility that something like the tolerance setting is set to .1" instead of .001")
2) The post processor is not correct for the machine.
3) The X-Carve is not mechanically sound.
My experience with users that have light duty machines is that the problem usually resides in the mechanical end of the spectrum. Maybe it works fine with Easel because it is not asking too much of the machine, where you may be setting your feeds/speeds in excess of what the X-Carve can actually handle. Maybe the times you have used Easel were under different conditions from those you tried using G-code from Fusion (Noise from something in the room causing stepping problems, etc.). Maybe a set screw has loosened and you are getting some backlash.
If it were me, I'd comb through the machine assembly and be sure everything is right and tight. I'd then be sure that the programming in Fusion is apples to apples in terms of feeds and depth per cut, while also using a post processor optimized for your application. If you are still having trouble, I'd then go back and send some hand written g-code with test shapes to determine if there is a problem with how the g-code is being processed. If I understand right, Easel connects to the machine directly, but G-Code must be sent through a different application that works as a translator. If that's true, the main difference between using Easel and G-code really comes down to that translator. Assuming all the mecahnical stuff is right, and knowing that Fusion outputs perfectly accurate code, the translator seems to be the spot where there is the most unknown and where I would be most suspicious the problem is stemming from. It may need a very specific g-code to work right, including how many decimal places the code contains before it may actually start rounding numbers.