Modern use of "I should think" vs. "I would think" in speech
Solution 1:
When conjugating the verb "to be" in the "proper" old-fashioned way, it was "I should" and "you, he, and she would."
The terms mean the same thing; the only difference is whether they're being used first person, or second or third person. "Should" for first person is now considered more formal, and alien to some ears, but for some of us it's just proper.
That's why Tom Lehrer and I say, "I should like to introduce . . ." rather than "I would like to introduce."