"Break-before-make" SPDT Relay out of 2 SPST Relays
Study this circuit: -



There is a single input called PWM but it can be any on-off circuit from any old logic device. It produces two outputs and notice the little bit of deadband due to the RC network.

For your application you can invert the OR gate output to guarantee it never rises until the AND gate output has gone low for the delay incurred by the RC network AND, importantly the AND output will never go high until the (N)OR output has fallen to low for the same period.

1 input, two outputs with shoot-thru (aka deadband) protection.

As an aside I'd probably use schmitt trigger input logic gates.