Digital Nomads

For what it's worth, I have never had a landlord try to verify anything with my employer. In case they try to actually verify your income statement, it's more likely, at least where I have lived, that they will just ask to see bank statements or some other hard proof you have had that income coming in. Maybe in some areas it's more common for a landlord to actually contact your job, in which case yes, it would reveal to your employer that you likely spend a fair amount of time in another state. Technically just having a lease means nothing, you could just stay somewhere for a few months a year as a vacation home but still rent it year round. It's going to raise questions though, especially if you aren't high income. But that's only if your landlord actually tries to call your company. And I know if my job was contacted by some dude claiming he needed my pay data, they would tell him to get fucked. I would probably have to authorize that explicitly, so you'd at least know if they tried.

I say I'm self employed. Because I am. So I can sign it myself.

Yes it would blow your cover, so find another way

Bank statements or a less needy landlord

Your company’s HR can usually generate a letter just saying that you work there. Why would a landlord need more than that?