Yes you can have both. My daughter and my wife both did last year. The only thing you will encounter is your new work insurance becomes your primary insurance and medicaide your secondary. So if you need surgery and your deductable is say $1,000. Your primary insurance will bill you the $1,000.00. Medicaide will come in and pay that $1,000 for you and then your first is responsible for every thing else. For the most part you should never pay a thing between the two. Owe and you will no longer be elidgle for a plan under meidicade. You will be moved to what they call open medicaide where you are not on a plan through them. We had a $4,000 deductable on my work plan when my twins where born. Medicaide paid it then my primary picked up the rest.
If your wife needs insurance have you looked into Share of Cost medicaide? I don't know your income situation but that is also a great option many do not know about.
Its not a jeep if it dont have 4wd.