Physical exfoliants aren't really effective as they can't reach deep into the pores, so they won't help with blackheads. Some can also be abrasive and damage the skin.
Chemical exfoliation is still an option for sensitive skin, but you should start with low concentrations of the exfoliating ingredient (AHA/BHA 2% or less). I believe Mandelic acid is one of the gentler ones, otherwise there's Lactic, Salicylic, Glycolic acid etc.
Once a week should be perfectly fine and not excessive.
After exfoliating, use a gentle non fragranced moisturiser, avoid other actives in the same routine.