When you, as an Admin of a Page, use Facebook, there are two different mechanisms for choosing which "voice" you are using -- i.e., working as "the Page" or working as "yourself".
The first mechanism (which is the solution to your problem) involves the "Use Facebook as:" menu-items. Initially, when you log in to Facebook, you are "using Facebook as" yourself, as shown in this picture (from my account):
Instead, you can/should choose to "Use Facebook as:" the Page in question:
... after which your "voice", throughout your entire usage of Facebook, is now that of "the Page":
At this point, everything that you do on Facebook, even when you go and visit another Page, is being done in the name of "the Page" -- which is exactly what you want.
Keep in mind, though, that as "the Page", you will only be able to Share from the Wall of a personal account, but not Like or Comment. However, on the Wall of Page, you will be able to perform all three operations.
The second mechanism is relevant only while you "Use Facebook as:" yourself, in which case -- while viewing a specific Page for which you are an Admin -- you have the option to choose whether to perform "posting, commenting, and liking" as yourself or as that Page, as in the following two pictures:
However, even if you are "posting, commenting, and liking" as that Page, this only affects the operations on that specific Page! But since you are still officially "Using Facebook as:" yourself and not that Page, if you then go and visit another page, "posting, commenting, and liking" for that other page will be done in your personal "voice" and not in that of the previous Page.
So, to summarize, the solution to your problem is to use the first mechanism: i.e., choose the menu-item to formally "Use Facebook as:" the Page you want, instead of as yourself.