OK I understand your paranoia and feel for you.
The GUMMINT and Fed agents don' t crash in churches! They commandeer the Motel 6.
I've stayed in churches MANY times. Just act cordial and normal. Don't be an a-hole. Introduce yourself. Honestly explain your situation. I show my ID as a courtesy so the pastor knows that I am not an insane wierdo, though few pastors ever ask for it. But when you lay out your cards honestly this is always appreciated, regardless of the church denomination.
Offer to help with their normal shelter routine. Especially if you have specialized training as shelter manager, EMT, LE .
After tornadoes, hurricanes or wildfires you are probably not the only one being sheltered. Offer advice and assistance based upon your training and experience. Help with intake, in the kitchen, triage of sick or injured, serve meals, wash dishes, sweep and mop floors. Hold hands and pray with them to comfort and console the hurting, if that's all you are OK with.
I always keep my weapon concealed and discreet so as not to "scare the natives." There have been a few times when I had to intervene to help de-escalate an EDP situation. Very common under disaster stress. Command presence works. The great majority of the time de-escalation can be done verbally and accomplished without going hands on. Only once did I need to cuff anyone. Once the cuffs came out I had multiple helpers jump in to help restrain the woman and prevent injury. It was just a panic attack and nothing threatening. Could have easily gone bad had anyone over-reacted.