what to do next..

There is a sticky at the top of this forum called the Primer at gives all of the information that you need for working in the US and what is required.

Please be aware that you can find an employer now and start the petitioning process, however, you are still looking at a few year wait for even a chance at getting a visa for the US, there is no longer any fairly good guarantee that all that pass the NCLEX exam will be able to get a green card. Number of applicants far exceeds the number of visas per year.

And by engaged in a hospital, do you mean that you have been hired by them or are currently working there? That term is not used in the US for talking about working somewhere but for being engaged to be married, or the telephone line is engaged because it is busy, etc. But we never use it to speak of a work related relationship with a facility. Justy pointing this out to you as you prepare for your English exams.

errm thought you had already passed NCLEX. Petition can not be started unless you have passed either NCLEX or CGFNS exam and to be honest employers would prefer a pass in NCLEX before offering petition especially now with retrogression plus a lot of hospitals are delaying taking on foreign nurses at the moment. If you mean before taking English exam then yes the petition can start before you pass it

It matters not what dictionary's say not everyone uses them plus I would say engage as in work is very old fashioned and I know in the UK is not used, employed is the word to use. I would say for English exams you need to be clear on what you are saying and listening to, many words have many meanings and you need to be clear on which one you use.