The only other "recent" state to adopt RTW was Oklahoma. So you can't really test that hypothesis by looking at changes in individual states. You have to do it by comparing RTW and non-RTW states, but because that's an inherently less reliable comparison, you have to be very careful to control for the many, many confounding factors that also vary between and among RTW and non-RTW states. In 2011, EPI did such a study
; they carefully controlled for over 40 other social and economic factors, and examined residual differences in wages (and other variables):
Wages in right-to-work states are 3.2% lower than those in non-RTW states, after controlling for a full complement of individual demographic and socio-economic variables as well as state macroeconomic indicators. Using the average wage in non-RTW states as the base ($22.11), the average full-time, full-year worker in an RTW state makes about $1,500 less annually than a similar worker in a non-RTW state.
The rate of employer-sponsored health insurance (ESI) is 2.6 percentage points lower in RTW states compared with non-RTW states, after controlling for individual, job, and state-level characteristics. If workers in non-RTW states were to receive ESI at this lower rate, 2 million fewer workers nationally would be covered.
The rate of employer-sponsored pensions is 4.8 percentage points lower in RTW states, using the full complement of control variables in our regression model. If workers in non-RTW states were to receive pensions at this lower rate, 3.8 million fewer workers nationally would have pensions.
This briefing paper provides the most comprehensive study to date of the relationship between RTW status and compensation. Using a full set of explanatory variables, including state-level controls, it is clear that our analysis stands apart as being more rigorous than others of this type.
Our results apply not just to union members, but to all employees in a state… We measure the particular effects of RTW laws on compensation among workers who are not unionized or covered by union contracts. The wage penalty for nonunionized workers is 3.0%, and the benefit penalty is 2.8 percentage points and 5.3 percentage points for health and pension benefits, respectively. Our results suggest that proposals to advance RTW laws likely come at the expense of workers’ wages and benefits, both within and outside of unions.
Note that latter point: unions don't just raise wages for their own workers; even non-unionized workers in states with above-average unionization rates receive higher wages.
Note also that cost of living was among the other factors that the EPI study controlled for (see table 4).
In contrast, studies that have shown economic benefits (on growth rates, etc.) from de-unionization typically only do a naive comparison of RTW vs non-RTW states without any statistical controls. That's good for propaganda, not so good for analysis.
The EPI study has the advantage of being comprehensive, thorough, and well-designed from a statistical standpoint. It has the disadvantage of being funded by, and published by, a generally left-leaning think tank. The latter concern is somewhat ameliorated by the fact that other "nonpartisan" studies have found similar results (e.g., here