Nenya or the Elessar?

[Galin]

Hmm, as you note (letter 131 states that) the chief power 'of all the rings alike' was the preservation or slowing of decay, and it's also said that the Three were supremely beautiful and powerful and 'directed to the preservation of beauty'.

I've not yet looked into the further distinction you wonder about, although I don't remember a 'stop' versus 'slow' distinction at the moment -- in specific comparison between the Three and other rings in any event.

There wasn't even a mention of her usage of Nenya until she was officially elected to rule Lorien.

There is a general mention that might imply the Three were in use earlier in the Third Age (Appendix B), as it is noted that for long the Elves were at peace, wielding the Three while Sauron slept -- this could be a general statement but my guess is that Artanis would have used Nenya before settling in Lorien (not that you said otherwise).

So do you think she was using the Elessar during her settlement at Dol Amroth, brief visits to Lorien, and her stay at Rivendell?

In my opinion Galadriel's settlement at Dol Amroth was rejected and essentially replaced with Lindon, although admittedly the evidence is somewhat thin. Concerning Nenya (and after the One had been lost to Sauron of course), to me it seems possible that Galadriel used it to help introduce the Mallorns before she settled permanently in Lorien. Cerin Amroth had Mallorns of great height, assuming they were planted there when the mound was raised, anyway.

And do you also think that she beautified Lorien with the Elessar instead of Nenya, for Nenya's main funtion was only to arrest decay and change. And it is said that all things grew fair about her with the Elessar.

I think the same expression could be used of Galadriel wielding Nenya, that all things grew fair about her (you didn't comment on that point so I'm not sure whether you agree or not, or that I made clear my opinion).