keep in mind that despite it being unwieldy and terrible to manage, the more dice you roll the more powerful you as a player feel. could apply to cards as well. imagine putting your entire hand on the table and saying 'I PLAY ALL OF THEM'
This sounds like a lovely game - I would play this.
Have you ever played "Freedom Force" on the pc? If you haven't - download it like an illegal and hideous pirate, or preferabley go and buy it from Steam (from me in the UK its currently £2.99) or come round my house and borrow my copy and play the damned thing.
If someone can make a game like that - they would be my hero
Edit: It's exactly the kind of thing you are describing - so yeah - your game sounds good
I think that a "maximum radiation reached" could be added. For example: - at max level, your hero gains a mega bonus, but will have to recover losing a couple of turns and restarting from zero - he could have to remove radiation lvl reaching an "hospital" - he reaches max level -> he dies