With a 30-year old rear axle, it's very possible that the limited slip is somewhat worn and sensitive to lubricant.
Generally, GM's lubricants are good choices but...it is pure BS that you must use only a GM lubricant and limited slip additive in a Corvette rear axle.
The recommended lubricant for a 79 axle was an SAE90 GL5-rated gear lube. You can get that from any source. Lubricant upgrades, such as the multi-vis synthetic you got from Castrol, provided they're compatible with limited slip diffs., can be a good upgrade.
As for the additive...more properly a "friction modifier"...
Some limited slips will make noise or will chatter. I'd expect that from the axle in a 79 which may never have had a lubricant change and may have had some friction modifier in the lube you drained out. A friction modifier works by reducing the friction between the limited slip clutch plates in the interest of reducing noise or chatter. The problem with this can be that you are actually reducing the very characteristic (friction) that makes the limited slip work. There is fine line between using too much additive and degrading limited slip action and using just enough to stop the noise.
The first step here is to determine of the Castrol gear lube you put in the axle is designed for limited slip differentials. If it is, I'd go to GM and buy one bottle of its limited slip additive. Add half the bottle's contents then road test. If it still makes the noise, add the rest and road test, again.
If the Castrol is not for use with limited slips, you'll have to drain it. Then, I'd look for any name brand GL5-rated gear lube designed for use with clutch-type limited slip axles. It doesn't need to be synthetic unless you are racing the car. Put the new lube in and road test.
If it's noisy, you need a bottle of GM's limited slip additive. Again, add half the bottle and road test, then add the rest of it's still noisy. If the noise persists after adding a full bottle, you can use more if you want, but what you really will need is a rear axle overhaul.