Whether there were any specific attempts to see why messages from Moria had stopped we don't know (as has been observed). As far as Golin's embassy, however, consider:
- Dwarves are very proud and don't tend to like to ask anyone else for help (especially Elves - even ones they are friendly with). That Dain was finally driven to seek Elrond's ADVICE (as in, "we dwarves are NOT wise enough to figure this out ourselves") is a big clue as to how IMPORTANT this trip was.
- While Dain didn't know what had happened in Moria, he had to suspect foul play. It's hard to imagine any other likely reason for Balin's new realm to go silent. PLUS, Dain knew that Durin's Bane had been there for hundreds of years (it was actually much longer, but Dain may not have known that) and had been single-handedly responsible for driving out a much stronger Dwarvish people than Balin had.
- "Checking On" Moria would have meant more than just waling into Dimrill Dale and gazing at the gates for a while. It would have meant walking INTO Moria through the Front Gate - where they are most likely to be seen if there ARE any Orcs there. IE, if you want to reconnoiter, it'd better be a reconnoiter IN STRENGTH - and even that is chancy.
- Remember, the Fellowship got in from the back door - where the Orcs were less vigilant. Also, they had a Wizard along who was able to take the Balrog out of the mix. Coming in through the Front Door just could not be done secretly.
- And if you DO try to reconnoiter and Durin's Bane (or Orcs) do get you, then you can kiss your embassy to Elrond Bye-Bye.
And that last point is why I think using Gloin's embassy for a scouting trip to Moria would have been foolish. Send another group, maybe - but if asking Elrond's advice is so important (how often to DWARVES ask for advice) then don't risk the Embassy on such a perilous side-trip.