While I can't explicitly tell you what you need for your evaporative air cooler I can provide a bit of information that you might find useful.
A couple of popular misconceptions:
First - Safety testing is not Federally mandated. It is voluntary. But the catch is that most retailers and distributors will not buy your product if it has not undergone some safety testing (for liability reasons)
Second - UL is not the standards body. They are a testing lab and safety mark. OSHA is the regulatory body that sets safety guidelines. The agency has approved 17 Nationally Recognized Testing Laboratories under the NRTL program
Each lab carries it's own mark. UL (Underwriters Laboratory) is simply the most universally recognized mark in the U.S.
Intertek for instance carries the "ETL" mark and is equally qualified and recognized by OSHA , insurance companies, etc.
With Electronic and Electrical products, OSHA collaborates with NFPA (National Fire Protection Agency) and references or abides by the NEC (National Electric Code) to set testing standards for safety.
If your product includes any type of radio transmitter (WiFi, Bluetooth, etc), your are REQUIRED to obtain FCC approval. I won't get into specifics here.
IMPORTANT: Safety testing and regulatory compliance for FCC kills many HW startups. Testing labs interpret the requirements differently, they often use slightly different tests, or focus on one area more than others. If your product fails a specific test that requires you to modify your tooling or re-spin PCBs you could lose months of time and many startups do not account for that potential burn. Preparation is the key to passing any test!
My advice is this:
1) Find a regulatory consultant as early in the development cycle of your product as possible. Have him/her review your designs, material selection etc.
2) Choose your testing lab carefully - Find one that has experience in evaporative air coolers. Trust me here.
3) Engage with the testing lab early. Get the actual test procedure they will conduct, step by step. Ask questions about each step so you know exactly what to expect and prepare for. You are paying this vendor, they are for profit, you are the customer.
4) Submit your electronic design and material selection to them (ie...plastics you are using for your mechanicals) to verify they it meets spec for flammability etc. in advance.
5) Submit for "Pre-certs". Pre-certification tests are practice tests. You can even submit for these tests using CNC'd plastics (as long as it is the same material you intend on using) and pre-production units. It will cost extra, but wouldn't you like to know if you are going to fail ahead of time while you are still able to make changes? In some cases the testing lab will actually count a pre-cert toward your final if you pass it and don't make material changes.
Happy to offer any additional help.