If states so strongly feel the need to execute someone, why go through all these hoops to pretend to make it look "humane?" Just use a firing squad and be honest (and actually effective) about what you're doing.
Death sentences make no sense anyway, they cost more than indefinite incarceration.
I'll share this, which should be pretty comprehensive: https://www.ojp.gov/ncjrs/virtual-library/abstracts/capital-...
But this provides more recent aggregate data citations: https://www.amnestyusa.org/issues/death-penalty/death-penalt...
Usually in these types of discussions, the cost being more is finally accepted. Then supporters of capital punishment move onto it being an effective deterrent, which is then easily debunked.