Peat can definitely be an acquired taste. You didn't know it before sipping the Laphroiag 10 but that distillery is considered by some to be peatier than most other expressions known for a smokey/peat taste profile.
The good news for you is that most scotch does not feature a peaty pallet and there are plenty of bottles for you to try without that seaweed smokey flavor that you detest.
So which distilleries are known for peat? Ardbeg, Laphroaig, Caol Ila, Lagavulin and Talisker are at the forefront but there are others. My guess is that somewhere less than 20% of the scotch whisky out there can be lumped into the "peaty" category.
Because being peaty is becoming more popular with scotch enthusiasts you should be able to find some mention of it on the label or in the packaging so I recommend you do some reading before purchasing. If you can, attend some scotch tastings. This will greatly improve your whisky experience and will lead you to other bottles that are similar to Glenmorangie.
If you like Glenmorangie I recommend something from Cragganmore, Dalwhinnie, or Balvenie. All are relatively easy to find and should be similar to what you have tried already.