Partition your graph into vertices of even degree, $N$, and vertices of odd degree, $D$. If the size of both sets is even, then you can add a matching to the set with lower degree to get a regular graph. If there is an odd number of vertices in one set, then it must be $N$ (and thus they can't both have odd cardinality). In this case, if the degrees are $2k-1$ and $2k$, so that the odd degree is smaller than the even degree, then you can add a matching to $D$, since it has an even number of vertices. If the odd degree is greater, you cannot add a matching to $N$, since it has odd cardinality, so you will have to remove a matching from the induced subgraph generated by $D$. It's easy to check if this exists, and that classifies your obvious graphs.
${\bf **Edit**}$ This construction fails to make simple graphs, as Felix points outs. It also can't identify which obvious graphs make simple regular graphs: e.g. {the graph made from $K_4$ minus an edge, $K_6$ minus a matching, and $6$ edges between them, one from each vertex in $K_6$ to give the other vertices degree $4$} is an obvious almost regular graph, but adding a matching to the $K_4$ part won't make a simple graph - you need to remove a matching from the $K_6$ part.