Avoiding fake trinitite
You will probably never know for sure, but the only way to get any confidence is to have any potential specimens analyzed by a mineralogical laboratory that can do Gamma spectroscopy.
Fake trinitite,

use a variety of means to achieve the glassy green silica look as well as mild radioactivity; however, only trinitite from a nuclear explosion will contain certain neutron activation products that are not found in naturally radioactive ores and minerals. Gamma spectroscopy can narrow down the potential nuclear explosions from which the material formed.

Also,

If it's too green, too translucent or too inexpensive to be true, it probably isn't real Trinitite!  So how does the collector know if a specimen is authentic?  Well, usually the only means for the collector to eliminate the fake stuff is to use a sensitive dosimeter and take a reading to verify that residual induced radiation exists.  This will be very small, but it can be detected.  This will not rule out the possibility that clever fakers will not add some small amount of weakly radioactive material to their formula.  The only way to be 100% certain of authenticity is by means of nuclear energy spectroscopy.

One of the radioactive elements to test for is Cesium 137,

This radionuclide has a half life of 30.2 years and is one of several unique byproducts of nuclear fission or atomic detonations.  It is present in all authentic Trinitite specimens.

All radioactive elements have a half life. The radioactive elements produced by the Trinity Test on 16 July 1945 are also well know. To increase the confidence of testing, the amount of each radionuclide in each sample would need to be detected and compared with the expected amount that would be in true Trinitite given the elapsed time since 16 July 1945.
If the specimens have a report from testing laboratory, get a copy of the report and verify it with the laboratory. Ideally, such reports should had a picture of the sample that was tested. If so, ensure it matches what is being offered for sale.