If your DM would allow it, you could scale down all the druid weapon proficiencies. For example, instead of being proficient with the scimitar (1d6 damage), you would be proficient with the small scimitar (1d4 damage), and so on. This keeps the weapons the same relative size for your small PC, and scales down the damage. There's a chart for scaling weapon damage up and down, but I haven't been able to find it. Try the DMG, and S&F if you have it.
Using the weapons you mentioned, a medium quarterstaff would be 1d4/1d4, and a medium shortspear would do 1d6 damage.
Since you are a small character, it is logical to think that you would have trained with smaller versions of the weapons. Of course, as a consequence of this it would be fair for the DM to rule that you aren't proficient with normal-sized weapons that you would normally be proficient with.