Mass Communication & Media

Yes ! you are eligible at most universities to take admission in Journalism courses you may wish to contact a college which provides good exposure of industry. look for colleges offering you placement and scholarship. I am not right person to talk about government institutes in journalism you may want to look at courses at
The Indian Institute of Journalism & New Media,
Mudra Institute of Communications,
Lovely Professional University,
Asian College of Journalism,
Symbiosis Institute of Mass Communication,
Indian Institute of Mass Communication,
Xavier Institute of Communication
etc (FYI. The list is based on internet research choose cautiously)

It depends on what stream you had in B. E. Like of you had core streams like civil or mechanical you can opt for govt exams. If you had IT then you should go for mba in any specialization of your choice as there are a lot of IT companies nowadays. Journalism should not be an option for an student of technical background, unless you are attracted towards it.
There is another option, an mba in construction or infrastructure Management for civil or mechanical graduates. This is a new field and has quite much potential as real estate and infrastructure sector is the fastest growing sector in India after fmcg. You can opt for this if you like. But there are only a few colleges that are good in this course. You need to choose wisely.
All the best.