The problem does sound like the RCM, you can remove it from its case as you have done before, inspect the relays for burnt contacts and missed bad solder joints. You can also put all the connectors back on it before putting it back in its case and see if it is operating properly. The air pump is normally the relay that gets burnt but it is not your problem, the ECM relay should be working all the time and the fuel pump relay as well.
Have you checked for any codes that may have been set? This should always be the first step, some codes do not set the engine problem light but as yours is lit all the time you would not know anyway.
If you need a new RCM they can be found new for about $135 new on eBay or auto parts suppliers.