"I should!"

I've been watching a few episodes of the BBC drama series "Foyle's War."   Its a decent show, but what interests me right now is just an expression that the main character, Christopher Foyle, often uses that I had never heard before.   It works like this:  someone will ask him if he thinks that they ought to ____, or if he wants them to ____, and he replies "I should!"

For example "Do you want me ask all the jewelery stores in town if they've seen this necklace before?"; "I should!"  or "Do you think I should check the oil in my car?"; "I should!".

Americans would, in a similar situation say "I would," which is short for, I take it, "If I were you I would do that."    How to get from one expression to the other is obvious:  you drop the "If I were you," and you drop the _proverb_ "do that."  "Do that," is a proverb because its anaphoric for the verb from the previous sentence, e.g. "I would do that" is anaphoric for "I would ask all the jewelry stores in town."  

So, first question:  is this use of "I should," still idiomatic in British English?  Or have y'all degenerated to "I would" too?    

I say "degenerated" because it's pretty clear that what's being asserted is not just a subjunctive conditional--what I _would_ do if I were you--but a subjunctive conditional about a normative claim--what normative facts would obtain if I were you.

So the second question is, what is Foyle's "I should" actually short for.  I take it is something like:

"If I were you, it would be the case that I should call all the jewelry stores."  (1)

But notice that you can't even express that without the awkward "it would be the case that."      If you try to say "I would should do that," it becomes unparsable and horribly awkward.   "Should" doesn't like being embedded in a separate modal, and in fact even in (1) the brain recoils at the embedding of the "should" inside the "would."    I don't think our brains are at all comfortable with the very form of the construction.

And yet, when Foyle says "_I_ should!"  with the right emphasis on "<b>I</b>" we have no problem understanding what he means, not even if we are Americans who have never heard that idiom before.

I have some ideas about what this kind of example shows about language but they are fairly heretical, so I would hold off on expressing them for now. "_You_ should."