Generally, the fact that the data failed the Shapiro Wilk test doesn't say it is skewed. A symmetric distribution may not distribute normally and fail the test ...
Even if the sample data is skewed, it doesn't say the data is skewed, for this, you need to test the data for skewness.
For very big sample the Shapiro-Wilk test may fail normality even for a quite normal data (because it will find a minor change from the normal which will be significant because of the big sample.
Can you please show the histogram (or better paste the data) of the skewed group?
The following online will check for Shapiro Wilk test, but will also check the skewness of the sample and test the significance of the skewness.
http://www.statskingdom.com/320ShapiroWilk.html
But if you just create a histogram you can check the chart, and if it is reasonably symmetric you may use the t-test.
Please notice, unlike the t-test that compares the groups' averages, the rank test compares the entire distributions. (and this is not a problem)
When the two groups' distributions have a similar shape, the test will also compare the median of each group.
For a symmetric distribution, the median is the average