Working part-time after graduation

I graduated from vocational school this month and the company that I currently work part-time for is willing to hire me and sponsor my visa. However, it is taking some time to get the paperwork organized. I had concerns with the legality of continuing this work.

According to my teachers, I am a student until the end of March, but after March I cannot do any work whatsoever, be it part-time or full-time without the proper visa regardless of me applying for the visa change or not. Is this true? We are planning to apply for the visa change in the middle of this month, and according to information on Immigration's website, it takes at least 2 weeks to 1 month to complete the process. (That is the information I saw on there, but some of you might say 3 months). I was concerned because if this bleeds into April, would I still be allowed to work for the company part-time until the process is completed?

And for which visa will you apply, because you cannot apply for a working visa as long you do not have a sponsor and on a a short-term visa/visa waiver you are also not allowed to work. Student visa is also no option since you are not fulfilling the requirements of the visa