Well, like Mark said, if you’re using HEC as your sole emulsifier, that solves your stability problem right there - HEC is not an emulsifier.
If there’s nothing else in your formula for the NaOH to react with, and you added it at 70C or above, you’re probably right that some or all of your oils have been saponified, and that would certainly account for the improved stability, and yes, saponifying oils will certainly change the texture of your emulsion. To prove this, add the same amount of NaOH when your emulsion is at room temperature - at room temp. the saponification reaction takes weeks, if not months, to run to completion, so you won’t see an improvement in stability if saponification is what’s helping.
There is absolutely zero correlation between pH and stability in emulsions generally.
Without seeing your formula, the best thing I can suggest for adding stability without changing the texture much would be one of the Pemulens.
Sadly, the least stable emulsions usually feel the best on skin. There is probably no way around the fact that stabilizing your emulsion will alter the feel/texture of your final product.