Same recipe but sometimes white chocolate cookies come out strange
White chocolate has a higher fat content than milk chocolate.  That's very likely throwing off the balance in your dough.  Even though you're using the same batch, more fat is being added to the white chocolate cookies as the chips break down.

You may need to reduce the butter or margarine that you use for the white chocolate cookie dough by a bit.  It's probably a very fine difference, perhaps 10-25% less butter.  That's a bit of a guess but please let us know your results if you give this a try.