Had the notorious mouse problem with a Steam Install of Mass Effect on Crossover Linux (Ubuntu 13.10--yes I know it's old) but found a fix that works for me, so it might work for some others:
Open Crossover window and select "Run Command" on the bottle in question.
Type in regedit to open the Registry Editor.
Go to HKEY_Current_User/Software/Wine/DirectInput
Look at MouseWarpOverride and write down somewhere or take a screenshot of the current setting (in case you need to change it back for another game/application that's running in the same bottle).
Now double-click on MouseWarpOverride and change the Value Data to read "disable"
Launch Mass Effect, or whatever. I find the mouse is working. It isn't trapped in the window, but if I set my resolution well, it's not much of a problem. Seems to be working well enough for now. Hope this helps someone. I'll post an update if something changes.