I have 3 data sets and performed One way ANOVA to see if means between group are different. The ANOVA for the P-value is 0.13 and shows there is no significant difference between means, but the four test for equal variance test O'Brein, Brown-Forsyhe Levene and Barrlett have p-value equal to or less than 0.00723. This indicates variances are not equal. The Welch's ANOVA test also has a low p-value 0.0123. Since the variance are unequal is the Welch's Test is the appropriate test for showing at least one the group means is different from the others since variances are unequal? I believe the Tukey-Kramer test would have been the appropriate if varainace are equal. Is there an eqaulivalent test if variances are unequal?
First off, try to avoid using p values as a 'cliff' indicating 'signficance' but more as a measure of probability of getting a test statistic at least as large due solely to chance. Additionally, what do your eyes tell you about group means when looking at the Fit Y by X scatter plot? And most importantly can domain expertise help inform your ultimate decision? What are the risks (practical not statistical) for making a wrong decision regarding groups? These should help guide your ultimate decision AT least as much as a p value.
Lastly, if you are bound and determined to use something that produces p values or their brethren in a multiple comparison mode, maybe take a look at the nonparametric multiple comparison tests? These are offered as a hot spot option from the Fit Y by X platform report under Nonparametric -> Nonparametric multiple comparisons.