There are two very different goals intertwined here:
One is distributing the load over multiples machines. You don't need to break out your application into multiple apps for that. You could use a load balancer for this (incoming requests are spread across multiple machines running the same app), or split your app up.
The other is more about the logical architecture of your app. In general the bigger an app is, the more complicated it is and it's easier to work on simple apps. Therefore you might want to split your app up into smaller apps. This is orthogonal to the hardware issue - you could choose to split your app up threeways but still run them on the same machine (it does of course give you the flexibility to allocate resources differently to your app)
Coming back to you question, one aspect worth exploring is should there be any shared models? To what extent can (for example) your dashboard query your api (possibly private apis) rather than querying the database directly? Splitting up your application into multiple services makes some things easier, some things harder, but it does help you keep individual applications smaller and keep the boundaries between them clean without any hidden dependencies. This logical split doesn't have to correspond the the physical layout: one server may host several applications
Lastly the most direct answer to your question is to write a rails engine. An engine can contain pretty much anything an app can (applications are special cases of engines), for example models, views, controllers routes etc. You'd create one with your common code, package this up as a gem (doesn't have to be a public one) and add it each app's Gemfile.