Relay to electronic switch change
You can use a depletion-mode MOSFET for your NC output, and an enhancement-mode MOSFET for your NO output. Enhancement-mode MOSFETs turn on when |Vgs| (the absolute value of Vgs) exceeds a certain threshold. Depletion-mode MOSFETs turn on when |Vgs| is below a certain threshold.
Since you don't mention voltage or current levels, I can't recommend a specific one, but take this one just as an example.
https://ixapps.ixys.com/DataSheet/DS100178A(IXTY-TA-TP08N50D2).pdf
You can search for &quot;depletion MOSFET&quot; in Digikey, Farnell, etc.