The F-statistic and p-value will not conflict with each other. The p-value is a measure of how extreme the F-statistic is – it’s a tail probability from the F distribution.

So, for example, suppose your criterion of choosing the alternative hypothesis is a p-value less than or equal to 0.05 (a commonly used value) and the F distribution critical point corresponding to this p-value is 2.2

Then if the F statistic is > 2.2 the null hypothesis is rejected. If this occurs, the p-value must be < 0.05

Correspondingly, if the p-value is < 0.05, then the F statistic must be > 2.2