overdispersion and ANOVA

Hello everybody,

I urgently need help with the R script for my Master thesis.
I have the following mod:
mod <- glmer(cbind(round(data$Time_cue), round(data$Time_ctl))~Task + (1|Colony), family = "binomial", data=data)
Cause of high overdispersion, I then removed colony and changed family from binomial to quasibinomial.
When using ANOVA, the p-value was 0.73. But when I then tested individually for my two different test groups, the p-value of group A was 0.043 and of group B 0.09 which is contrary to the ANOVA.An other example was ANOVA: p-value = 0.11, but then p-value group A: 0.96 and p-value group B:0.026
What is wrong? Is it cause the overdispersion is very high, do I need to change something else?
I would be very happy if someone could help me please!!

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