The transfer between terminals may happen automatically for your hold luggage, but you will have to go to Concourse A apron level to take the bus from T3 to T2 in order to get yourself there.
The free showers are only in Terminal 3 Concourses B and C. There are only paid showers in Concourse A. If you have enough time to go to one of them to take a shower before you go to the bus to change terminals, then there is nothing to stop you.
So it really all depends on how much time you have between your flights. Note that the minimum connection time is 120 minutes for an Emirates-FlyDubai T3-T2 transfer. So in addition to that you will need the additional time it takes to go to one of those shower areas and take your shower and return. I'd consider it safe if you have three hours between your flights and don't make any other stops.