Electric ground from another circuit
I would avoid using the baseboard ground for the outlets since the baseboard is on a different breaker (circuit). Per NEC:

NEC 300.3(B) All conductors of the same circuit, including grounding and bonding conductors shall be contained in the same raceway, cable, or trench.

You can certianly replace the home run and associated branches with 3/12 (20 A) or 3/14 (15 A) depending on the circuit breaker size.