That's normal, as you need to log out of the first Google+ account before you can log in to the 2nd.

It has to do with a cookie that Google sets at login, not the URL of a bookmark; the cookie overrides the URL.

How can I keep this from happening?

You can only be logged in one Gmail account at the same time. However, there is an extension which allows you using more accounts. Please consider trying it.

Okay, here it is - I have 2 bookmarks in my Email folder: One is for my personal Gmail account, one is for my school account. When I click on my personal account, it goes to the inbox. Great. I close Gmail. Two hours later, I click on my school gmail bookmark. It opens the last account I had open, my personal account. It will not go to my school account, which has a totally and completely different URL. It opens my personal account. If I have last been on my school account and close it, then later try to use my bookmark to go to my personal account, it goes to my school account, i.e. the last account I opened. The funny thing is, I deleted cookies, replaced bookmarks, etc., and for now it is working fine, but in a day or two will start sending me to the last account instead of to the clearly different address for the bookmark I am clicking.

This is a limitation of gmail, they only allow you to be signed into one account at a time, you can't use both, bookmarks won't work. Did you try the add-on above?