Should I apply for J1 & J2 together or separately?

#1
I am planning to apply for J1 visa to do part of my work in USA as a visiting scholar and it is completely funded. My tenure is from January 1, 2019 to January 1, 2020. I am planning to go in January, 2019 and take my Wife there around March, 2019. It would be good if anyone could let me know if it is possible to apply for a visa together, because our dates of travel are different? I would also appreciate it if anyone could let me know if there is any problem to apply for my Wife after I receive J1 visa? I am really confused which way should I go? Should we apply together or separately? Please suggest.
 
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