No, you don't continue the relationship because you have no relationship to continue.

First of all, do not blame yourself for what you said in the airport. You said nothing wrong, nothing that was unreasonable and he is totally gaslighting you on that.

Everything he has said to you has been a lie. Do you actually know, as a proven fact, that his mother died? Are you sure?

Do you know 100% he is who he claims to be and hasn't catfished you?

Either way, it's time to cut your losses. There is nothing wrong with meeting and starting as a LDR but you need to meet. 18 months down the line he has done everything he can to avoid doing that, he has gaslit you, demonstrably lied to you if you ever tried to meet him again, you already know it would be exactly the same.

The fact that he blocked you, his girlfriend, on ANY messaging platform is, IMO, unforgivable and an automatic breakup.

No, you don't show up in Hawaii. He won't be there. Go to Chicago, enjoy your holiday, go home a single woman and move on to dating real people