This addresses Windows Mail - not WLM (as that was the original question).
To my knowledge, there is no "solution" to this problem. That's how the programs work - even if it's obviously not how things should work. There's no setting or anything like that to change the signature from the default user to a different user when you switch users in the create mail window. I'm afraid when you use a different user and want a different signature (the signature for that user), you're going to have to delete the default signature and add the signature you want. But there is an easier way than just cutting the wrong signature and typing in the correct signature.
The easiest way to do this is to add multiple signatures in Tools / Options / Signatures. Create a signature and then click on advanced and choose which account you want that signature to apply to. Then create the other signature and click on advanced and choose the account to which you want that signature to apply. Make sure you name them properly so you'll know which is which.
Now when you create a new mail message it will start with the default user. You then change it to the other user but the signature will not change. Highlight the signature and go to Insert / Signature and select the signature for this other account and click on it. It will overwrite the default signature and insert the signature for the other account. It's not perfect and not the way it should work, but as far as I know it's the best you're going to be able to do.
Why they didn't take this the one step further and have the signatures automatically change is beyond me - but unfortunately, that's how it works and what I just presented is the best workaround I know.
I hope this helps.
Good luck!
