There seems to be a little confusion.
1) Levene's and SW are tests of homogeneity of variance which is one of the assumptions of ANOVA. One test showed non-homogenous variance and the other did not.
2) Games-Howell is a test of equality of means when the assumptions of ANOVA are rejected. I had not seen it used before, but it appears to be fairly widely used. So, you can probably use it. But you might want to do more research into it, particularly if this analysis is highly critical.
3) You didn't report an F test for the ANOVA itself. That would tell you whether there are differences among the means (but it makes those assumptions). Some people think that you should not do post hoc tests if the ANOVA is not significant, particularly if particular differences were not specified a priori.
3) Alternatives to using Games Howell include the methods in @Steven 's answer. However, it is hard to say which approach is best without a lot more information about both your study and your results. Even if you conclude that the variances of the residuals were different - why and how were they different?
My blog post "How to ask a statistics question" may be of help.