This data is taken from the 1969 Chrysler Corporation Mopar parts catalog page 17-1.
Barracuda 318 and 340 Manual and automatic transmission use part number 3004588, I was surprised there wasn't a special 340 spring. No other A-body used this number in 1969.
Barracuda and Dart 383 Manual and automatic transmission use part number 2808677. No other A-body used this number in 1969, which didn't surprise me at all.
In both the above cases there isn't a left and right, the same spring is used on both sides.
There are two section headings for springs, Standard, and Heavy duty. Both sections list the exact same part numbers for these applications so these are the only two numbers. I don't know what the difference is as I have not seen them side by side. Factory 383 A-body's were uncommon enough I never did see one in the Mopar dealerships I worked at from 1970-1978 so the opportunity to compare them never presented itself.
If I had to wager I'd expect the 383 had an extra front half leaf at the bottom of the stack, or an extra short leaf just above the half leaf.
All of those cars were equipped with F70-14 tires. I'll bet the 383 cars melted those in a hurry