It doesn't change the base class's wording, but there is a separate feature that states the changes.

This should absolutely change what the base class says on the sheet. The sheet is a particular instance of the rules and all the text should be in context of that character, not general rules.

Since most of the Circle of the Moon's focus is on Wild Shape, it is quite crucial to get the correct info.

It would be handy to get the Wild Shapes in a more convenient location or even have a separate sheet per shape so we could easily roll our attacks/saves and skills from those pages.

I also noticed that the Insignia of Claws correctly adds the +1 for unarmed attacks, but doesn't change any of the attack values of the Wild Shapes, which it should.

The sheet will give you the base class feature - as it should - and details the features of the subclass separately, like it should. This is the normal way it is done. This shows the correct info so you can refer to what it needs to be without looking in the PHB. All you need to do is make a mental note to look at subclass bit ("Circle Forms") instead of base class bit. Simple enough. If you do struggle with that for some reason then you can make a custom action on the sheet named Wildshapes CR, set as bonus action put your wildshape CR for damage, enter notes for CR rule (1/3 Druid Level) and choose display as attack. In your Bonus Actions (since as a Moon Druid it's a bonus action) you'll see your current CR and a little further down the note of it being 1/3 druid level. You'll have to adjust whenever you level up, but that is there to remind you of the rule in a more prominent place.

You can use the Extras tab to add wildshapes to your sheet.

When you wildshape what you wear and carry - like your equipment and items - is either dropped to the floor or merged into form. In either case you no longer have it on you and in the case of the latter you no longer have access to it at all. The worn insignia of claws therefore wouldn't apply to your wildshapes because you're no longer wearing it. The insignia is a badge you attach to clothing. The system therefore doesn't support "wildshape clothes and items" due to the impracticality (you'd need to have somebody else come and it to you each time you wildshape). But, if you need it to apply then either just remember to add +1 to attack rolls or create copies of your most common wildshapes and edit to factor the +1, rename to make it reflect it, then add that homebrew version to your sheet.

Still better than pen and paper.

Quote from Entifidich >>

Since most of the Circle of the Moon's focus is on Wild Shape, it is quite crucial to get the correct info.

It would be handy to get the Wild Shapes in a more convenient location or even have a separate sheet per shape so we could easily roll our attacks/saves and skills from those pages.

I also noticed that the Insignia of Claws correctly adds the +1 for unarmed attacks, but doesn't change any of the attack values of the Wild Shapes, which it should.

 It cannot. The software cannot do that. The insignia of claws applies to the PC, but the software has no way of applying it to Wild Shapes since their attack and damage rolls are written in plain text and do not have any coded “actions” associated with them to populate the actions field of the character sheet. That’s why we cannot roll dice from the extras tab either. They are apparently working on a solution to that issue, but there are no ETAs.

I do not see Circle of the Moon available in the drop down list any more. Is it still an available selection?

Quote from NavariaForests >>

I do not see Circle of the Moon available in the drop down list any more. Is it still an available selection?

 Have you purchased Circle of the Moon or the PHB here?  If not, and you had access to Circle of the Moon before, you were almost certainly in a campaign with content sharing turned on. Content sharing only affects characters actually in the campaign.