Assuming that your data support the analyses that you've started...
The t-test really does depend on the normality assumption; when that fails the p-values it gives may be in error. This failure is often in the direction of a smaller p-value that is really supported by the data. So I would be suspicious of the tiny p you got from the t-test, especially given the much larger Wilcoxon number (I will assume that you set that up correctly.) I've seen the situation you describe in my own consulting work. What you must consider for yourself is how severe is the skew? If it is a lot, the t must be abandoned in this case I suspect.
Failing normality, the t-test works well for data that is unimodal (your description suggests this may be true) and symmetric about its center. Clearly this last fails as you said that the data you have is skewed. If the skew is a lot, as I said, the t-test is not really justified.
Also, as an aside, given your vastly different sample sizes, the classical Student t-test should be replaced with the SWS t-test (called Welch's t-test, Welch–Satterthwaite test, etc.) This test uses a different degrees of freedom calculation to deal with unequal sample sizes and (possibly) unequal variances. There are statisticians who believe that it should always be used in place of the classical Student t. (I'm one of them; sorry I don't have the reference handy!)
From your description the Wilcoxon test is a good option, but as it is fully non-parametric, you may be losing a lot of power to detect differences that are actually significant. Another option would be to go the resampling/bootstrapping route. Or to use approximate permutation testing, assuming the groups are independent (you did not say, but the Wilcoxon also assumes this). But you will have to do some research to figure out how to implement these tests in your software. If you want to follow this path, and you are using R, a good start is at this post. I think it might help you.
However, if the data is truly skewed, and you must pick Wilcoxon or t, I think you have to live with the Wilcoxon.