looks like the filling station has a rebuild available on the valve, if this is the same thing? https://store.fillingstation.com/detail/FS-299/Chevrolet_RANCO_HEATER_VALVE_REBUILD.html
a lot of places either don't have a '62 heater core listed or they have the heater core like I have in my '65
] listed as fitting them all, which I do not think is correct for your application, from looking at your photos.
I'll poke around a little more, see if I get lucky.
found a few listed, but not sure if they are the same as in your photos... judging from your photo angles and theirs, but check these out below. Good luck!
rockauto.com has several listed for a '62 chevy c10 truck, the ones toward the bottom may be like the one in your photos, but not sure. check it out. nevermind, out of stock, but check the photos anyway.
listed, but dunno if is it correct or not: https://www.cjponyparts.com/c10-heater-cores/c/5511/?product_year_c10=4213
ebay has several listed, check the details.
lmc is all backorderedhttps://www.classicindustries.com/product/3145850.html
advance auto parts shows them in their system/website(they may have to order it).https://www.npdlink.com/product/core-heater-copper-brass-9-12-x/194461/100664?year=1962
classicparts has two listed for 60-63: