Is this method secure?
From what you've stated you probably haven't opened up any security holes. The session cookie is not itself encrypted on the users machine, but you are making sure it is encrypted between their machines and yours, as well as between each of your machines. You should make sure you've set the Secure Flag to help prevent the cookie being accidentally sent over traditional unencrypted transport (HTTP), but as stated, this doesn't effect storing the cookie itself.
That being said, you are essentially hijacking your own users sessions. While a hole might not be introduced now, you are potentially weakening the overall system, which could lead to hole in the future.
Is there a better way to do it?
This might well be a dumb question, but are you sure you need the session? If you're juggling credentials between servers, it sounds more like you want to use Access Tokens and scrap the session.
Using Access Tokens is similar to using sessions, but you need to make your services stateless. This means your no longer storing information about the logged in user any specific machine so you'll need to pull anything you need from the database every time they hit a server requiring that information.
This is a good thing in the long run as it's much easier to scale your services when you don't need to worry so much about where the session is and what's inside it.
OAuth 2.0 is widely used standard (Facebook, Twitter, Google), and was specifically designed to be easy to use. The RFC is complex, but there's a log of good guides out there don't worry.
The one slight down side (if you can call it that) to OAuth 2, is that it MUST happen over an encrypted connection. If your use case can not guarantee encryption over SSL or (preferably) TLS, then you should use OAuth 1.0 (WITH revision A) instead.
This is due to the fact that OAuth 2.0 exposes it's "secret" token in requests, where as OAuth 1.0 only ever uses it to provide a signature hash. If you take this route it's advisable to use someone else's library as the hash is very, specific.
Further Improvement
(Note: This section added after the answer was accepted)
One system I've been exploring recently is Json Web Tokens. These store information about the user to save each machine repeatedly looking it up in a database. Because the token is hashed with a secret, you can be sure that, so long as your secret isn't exposed, a valid token represents a successfully logged in user, without having to touch the database.
You should avoid putting anything too personal in the tokens if possible. If you must store private or secret information in the token, you can encrypt it, or you can use a reverse caching proxy to exchange the JWT for a traditional security token. This may initially seem to defeat the purpose, but it means some of your services may not need database access at all.