Oman is awesome- have lived here some time now. Yes, check out the museums etc but the really special places are out there. Jebel Shams and Jebel Akhdar (4x4 needed), snake gorge is great to check out though check your weather as if it does rain the gorge will be out of action. Or do some standard 4x4 off roading up Wadi Bini AWf to see the gorge from an easier track (easy 4x4 here, beginner friendly).
I love going to Ras al Hadd to see the turtles (seasonal), it should be warm enough for the Daminyats also if you hire a boat for the day where you will see amazing things from whale sharks (occasionally) to rays and turtles.
The desert is worth it- Bidiya isn't that far and is amazing. It will be cooling down at this time so better to go. There are plenty of desert campsites down there that will do anything fromc amel rides to dune bashing. Some are even dog friendly (from experience).
I'd recommend a weekend to Salalah in the south also to see Ubar (the lost city) and the Queen of Sheba ruins as well as the last of the Khareef (october won't be khareef season but it may still be nice). Try a few wadis: Shab and Bani Khallid are tourist favourites, the sinkhole at Bimmah is near Shab so you can do a combi trip.
Mutrah Souq is the tourist must do- I actually prefer Fanja which is famous for it's pottery or going across to Seeb which is a bit more local and less tourist (smellier though!). If you want to relax, Jebel Sifah resort or the Shangri-la are a short hop on the highway and a nice excuse to chill out.
In regards to another country, the whole GCC is interesting, depending on your own country of residence. I've checked out Jordan recently also but it's not as close as you think (4 hour flight) but an amazing country to see. Many people hop back via Jordan, Lebanon etc on their return to the UK/US from Oman. The cultures are very different so you might appreciate different this more coming from Oman. Dubai is very cosmopolitan but at the sane time it can feel like it lacks culture as more expats live there than locals now. It really depends what you want to do. 3 months ago