I think the tactile feedback is a separate issue from your non-working carts.
7800 carts have a different card edge (compared to 2600 carts) which will change their feel. Also, many 2600 carts have a cart edge cover that the 7800 and later 2600 carts lack. Having that cover, or lack thereof, will change the way that cartridges feel when inserted.
My advice: Clean the cartridges & the pin connector on the console. Forget about the squish.