Sadly, I remained stumped as to when "which" or "that" should be used, other than the gut feeling of whether or not it sounds funny. Can anyone educate me on this? If you tell me, I will buy you drink.

The standard prescription requires the use of which with non-restrictive clauses and that with restrictive clauses.

Thus:

The house, which I bought last summer, has excellent access to the lake. This would refer to the only house I bought, or at least, the only house under discussion at the moment. I am only offering additional information about the house, i.e., that I bought it last summer.

The house that I bought last summer has excellent access to the lake. In this case I am allowing for the possibility that I have bought other houses or that there are other houses under discussion. The access to the lake from these possible other houses is not of concern.

In practice you will find this rule disobeyed all the time. The minor fact that the inclusion of commas in the non-restrictive example and their exclusion in the restrictive amounts to redundancy doesn't seem to bother the word doctors. In the end, go with the old rule: if it sounds funny the hell with it.

My wife attended Girls' Latin School in Boston, and is fond of referring to the " 'which' hunts" they were encouraged to go on, looking over their English papers before handing them in as completed...shouldn't the school have been in Salem, then?

Note the commas separating the "which" clause from the rest of the sentence in the post above, and the absence of same for the "that" clause.

A little redundancy now and then din't never hurt nobody.

Go with whatever sounds right. Fowler's Modern English Usage says something like the language would be a lot tidier if 'which' were used for restrictive and 'that' for non-restrictive clauses, but the fact is the language is not tidy.

Since Fowler's day, some people have tried to impose this system on the rest of us, but they haven't succeeded. File it with split infinitives and ending sentences with prepositions as one of those things the linguistically insecure get upset about for no real reason.

Bingley

So the question of which we should use, which or that, in any situation is something that is prone to hyper-flexibility. Thank God! The restrictive/non-restrictive rule (I hear you, musick! ) is pretty clear, but there're still times when one or the other just sounds awkward even if it's the correct usage grammatically.
And sometimes, neither sound right. And I could never rememebr if there was a grammatical edict that would help simplify the decision. So, yeah, what Faldage says...if it sounds funny, the hell with it.

Which that or which is it proper to use
When a which or that is that which you'll chose?