Crossman 760, so disappointing.

As for that 760 and the loose barrel, if it's like the 75 I worked on it's a standard barrel that is set back more and a notch and new transfer milled into it. If it rotates then the transfer probably isn't locking it in and may be worth checking out. The pump lever is a complaint I've seen a ton, but i bet you could retrofit it to be better if you were inclined.

I don't know if the probe is still a no oring deal but that upset me nearly as much as my refinish debacle that I wrote a safety psa for people on. Even in 75 it looks like it was essentially a "spare parts" mutt build from crosman. The valve is a peculiar design too, at least it was. I'd expect yours is a 13xx or 22xx style aluminum valve and the same pump piston as they've always been. I'm happy to chime in if you tear I to it and help where I can with my experience on a bunch of old guns i get for free and fix up.

Those Benjamin's, sheridans and later benjamin-Sheridan models, are pretty good options if they can pump them. It's the worst gun to suppress but it can be done. I hope your kids really enjoy the hobby.
 
I think I got my 760 in '67. IIRC, it had a BB reservoir and a 10-shot reserve. You pulled back on a little switch, pointed the rifle down and gave it a shake...10 more shots!

I was so proud when I could do 12 pumps (took me all summer to build the strength). I made a bike holster so I could ride further afield.

I was allowed to shoot anything but songbirds. Took Skunks, Bunnies and about a bazillion Sparrows and Starlings. I was the neighborhood sure-shot.
 
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