Can my ex-wife move 300 miles away with kids?

In a situation where a custody order is already in place, it cannot be maintained if one or both parties move out-of-state.  Accordingly, a move could mean that you are in contempt of a court order. Your wife would have to file for modification of the existing custody order to obtain the court's permission to move with the minor children.

The factors in determining if such a move will be allowed include, but are not limited to: the motivations of the parent desiring the relocation; the motivation of the parent opposing the move; the advantages of relocation in terms of its capacity to improve the life of the minor child; any disadvantages of relocation on the minor child; the likelihood that a reasonable visitation schedule can be arranged which will preserve and foster the parental relationship with the noncustodial parent; the likelihood that the parent desiring the move will comply with visitation orders after they have relocated to another state.

A parent who wishes to relocate because of a job, new spouse, or to care for sick relatives will have a stronger case than a parent who simply wants to move away from the non-custodial parent. Similarly, a parent who can demonstrate that the move offers numerous advantages in terms of its capacity to improve the life of the minor child has a stronger case than a parent of a minor child for whom the move offers numerous disadvantages. 

Considering her history of alcohol abuse and the fact that she would be leaving both the area that you and her parents live in, I think you can make a strong argument for having the move denied.  At this point, you should consult an attorney with experience handling these kinds of cases.

Best of luck.