lucia and jason should have separate money and separate wanted histories
Character switching only matters if the characters actually live different lives.
Lucia and Jason should not share one magical wallet. If Lucia earns money, buys a car and attracts police attention, that should belong to Lucia. Jason should not wake up across the state with the same cash, weapons and clean record.
Shared safehouse money can exist for major purchases. Personal money should handle clothes, vehicles, supplies and dumb impulse spending.
The same applies to police attention. One character using a recognizable car in repeated crimes should make that car risky for that character. Switching protagonists should create breathing room, not erase every consequence in the world.
Yes, this adds friction. That friction is what makes switching a meaningful decision instead of a teleport button with a different voice actor.