With the date
command line tool you can “normalize” your dates to Epoch time:
Your first format (with the time zone):
datestringOne="2016-05-01 10:02:32" # -0500
TZ="America/New_York" date -j -f "%Y-%m-%d %T" "$datestringOne" +%s
→ 1462111352
####Your second format:
datestringTwo="5/1/2016 15:02:32"
date -j -f "%m/%d/%Y %T" "$datestringTwo" +%s
→ 1462107752
(depending on your actual time zone)
To the second string I have appended the missing seconds. Otherwise the current seconds would be added by the program.