I think it is possible to work a system to integrate randomness before decision-making even in a traditional role-playing game, if that is what you want, especially for systems where the randomization is mostly based on rolling a single type of die, such as D&D and other d20 systems.
Basically, at the start of each adventuring day, you roll the die (say) three times. Each time you are supposed to roll dice to determine an outcome, you add the result of that roll to your pre-rolled results, and you may choose to use any of the four. The result you used is discarded, leaving you with three potential die rolls again.
In game, you can keep track of this by noting the die roll results on your character sheet and erasing or scratching them off as they are used, or you can use a pack of ordinary playing cards, less the face cards, with hearts and diamonds being 1-10 and clubs and spades being 11-20. The latter approach does not precisely mimic the probability distribution of separate d20 rolls, but produces more "balanced" results in the sense that bad initial draws make it more likely that you will draw good cards subsequently, and vice-versa.