The t test assuming unequal variances that most statistical softwares uses, uses the Aspin-Welch (WA) test. This test "estimates" the degrees of freedom. I would recommend you to use WA if you dont know that there are equal variances between groups, which you usually do not know since otherwise you wouldn't want to make hypothesis tests.
I would not recommend anyone to use the OTS for any real world problem, and the arguing for that is: i) if we are dealing with samples generated from populations with different variances, WA is a better choice of test; ii) if the samples are of unequal sample sizes, we are better off with WA; iii) if one is testing for equal variances in advance in order to decide whether OTS is appropriate, the power of this test is going to be low, meaning that the risk of falsely not rejecting the hypothesis that the variances are equal will be high, which leads to; iv) different nominal levels of the test we decide upon since we are testing for equal variances in advance; v) and the final argument is, if the OTS is a good choice of test, WA seems to be approximately equally good.
Google "Behrens-Fisher problem" if you're more interested in this problem.