Just wanted to add: the area is bordered by cedar trees, which love to shed their leaves/needles on my grass, and we get a lot of rain during the fall, winter and spring. I try to pick up the cedar leaves as soon as I can when they fall, but some still make it into the soil. Liming has helped, but it seems to be a short term thing, and I'd be willing to lime more often if it is ok for the grass/environment.

thx.

Whether you need, which kind you may need, and how much to properly add can only be determined by a good, reliable soil test. AgCanada does not do that but your local office may be able to provide you with the names of labs that do.
Whether any lime, Calcitic or Dolomitic, is organic or not depends on if it is crushed limestone or has been treated to be more readily available. Hydrated lime would not be organic.

I thought I replied to this but I must have not submitted it.

Plain lime is calcium carbonate which comes from limestone. Dolomitic lime is calcium magnesium carbonate which comes from limestone impregnated with dolomite. They both have similar power in adjusting pH but regular lime provides calcium while dolomitic lime provides calcium and magnesium. If your soil test doesn't indicate you need magnesium you don't need dolomitic lime.

Hydrated lime (calcium hydroxide) is quicklime (calcium oxide) that has been slaked with water. Quicklime is just calcium carbonate that has been heated in a kiln. That heating is enough for the NOSB and OMRI to consider it synthetic and not appropriate for use as an organic soil amendment.

The reason some people use hydrated lime is because it acts faster and has the ability to change the pH more.

There are products out there that get better results than using regular ground limestone. They do this by increasing the purity of what you spread (% of calcium carbonate in the bag) along with grinding it very fine and pelletized to make it easier to spread. There have been some brands of pelletized lime that have OMRI approval but I wouldn't worry to much about making sure if it had OMRI approval or not in this case. Not every company seeks OMRI approval, especially in cases where a product is used commonly in both organic and non organic applications. Just look for something ground fine, has a high concentration of calcium carbonate and is pelletized.

Stay away from hydrated lime, more because it can burn your lawn than whether it is organic or not in my opinion.