I should point out, 2a and 2b both would be more natural to me if you used "very" instead of "much" -- "much worried" is not typical.
I don't claim to be representative of all speakers, but the use of "last time" to replace "the last time" sounds okay in both of these examples. I think it's much better in the first one than the second though--while I might expect to see 2b in casual speech, it would seem like swallowing or skipping over the "the" rather than something I know I would write.
Some web examples of the usage include a Facebook group called "Last time I heard that, I laughed so hard I fell off my dinosaur!!!!!" (...for whatever that's worth as evidence) and the headline of this Yahoo! news article. I think that's enough to show that it's recognized as legitimate by the average US English speaker. It also seems to come up a lot in song lyrics. But it seems most natural to me if it's at the beginning of a sentence.
In conclusion: when in doubt, use "the", and you won't be wrong. But I think you're right that you can probably omit it in many cases.