F TEST measures the variance ratio of two samples, i.e. var1 / var2. Intra-Class Correlation (ICC) measures is the ratio of one var over the total of two variance" va1 / var1 + var2. The interpretation for each test may not be the same. With the results you have, you should use both and report each with its significance meaning separately.

LINK:

https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Confidence_Intervals_for_Intraclass_Correlation.pdf

I agree with reporting both values and their meaning, but the actual problem is that your values are too close to the cut-offs for nonsignificance. This means that your findings are barely significant and that they are very fragile to minor alterations like adding or subtracting one participant. Even if your findings are truly significant, these types of results usually signal low effect sizes. In such cases, therefore, it is better if the researchers are cautious in interpreting that they have rejected their null hypotheses.

A useful reading may be: https://journals.plos.org/plosmedicine/article/file?id=10.1371/journal.pmed.0020124&type=printable

I agree with the previous answers, but looking at the range of your 95% CI which is so wide, I advise you to inspect your data for outliers, your ICC is acceptable but the wide range of your 95% CI seems affected by an outlier, it worth having a general look at your entire data.

best wishes