Asymmetrical Bell-curve Roll, or ABR for short.
With ABR, you get a flat topped bell-curve. Basically, the middle half of possible values are each equally likely to occur, but the extreme low and extreme high values are reduced in likelihood.
In ABR, you roll 1d12 and 2d4 instead of a d20. the range of values is 3 - 20, with the average roll hovering around 11.
Because the upper limit of ABR is still 20, there is no need to alter crit ranges, though it might be prudent to add +1 or +2 to all crit ranges if you want critical hits to still happen nearly as often as normal.
Anytime a rule mentions rolling a 1, assume rolling a 3 instead.
Advantage and disadvantage.
I never really liked the idea of advantage being to roll twice and take best result, so here is an alternative that lessens the impact of dis/advantage, allows stacking multiple dis/advantage, and yet remains simple and easy to implement.
For each advantage roll an additional d4, then take the two best d4s.
Disadvantage works the same but in reverse. Roll a extra d4 for each disadvantage and take the worst pair.
If you have both advantages and disadvantages, have them cancel out one-for-one.
Alternatively, keep all advantage and disadvantage, rolling an extra d4 for each, then for each advantage, discard the lowest d4, and for each disadvantage discard the highest d4. At the end you will have 2d4 remaining plus the d12.