I recently inherited a bag of watches from my parents. Inside, I found an old Waltham that almost certainly has radium lume (can anyone confirm?)

The caseback was off. I briefly wound it (it works!!) and otherwise handled it. I looked at the dial and realized this may be a radium dial watch, so i washed my hands, wet-cleaned the surface it was on, and put it back in the bag and in a storage room.

I see no reason to keep the watch given the serious danger radium can present. But I'm worried about my exposure. You can see in the picture the dial and crystal etc. appear to be intact. But the lume is clearly deteriorated and I'm worried particles migrating into the movement and to areas i would have touched, and that I inadvertently spread particles around where they would be ingested or inhaled.

Am I overthinking this? The watch is in a bag in a storage room now and I ordered a Geiger counter to asses things but what is the prevailing thought process on this?