One of the interesting ways of adding a “handicap” without breaking the balance is to make some promises. So you would playing the standard rules, but in a way to make your life harder.
As a minor handicap you could say that you wouldn't discard any Destination Tickets on start. A bit more serious — when getting new Destination Tickets you would keep two instead of three, or even all of them.
Moreover, you could spend the first turn by getting more Destination Tickets without doing anything else. Or say you'd draw more Destination Tickets when you'll spend half of your train cars.
Also, you could reveal one or more Destination Tickets, so our opponents would know where to block you.
You could say you would always draw in a specific way, like always take two random cards without touching the face-up ones, or always draw one face-up and one random. Or don't ever draw locomotives from the face-ups.
Also, you could then show the drawn random card, or always show any number of cards in your hands.
You could say you would claim routes in a specific order. For example, you could say you would claim routes starting from the smallest and at any time you would claim the route that is either higher or smaller than previous by one. So, you could claim 1-2-1-2-3-4-3-4-5-4 etc. There could be many variations of such handicap, like messing with colors or saying you would claim only those routes whose colors are there in face-up cards.
All those things wouldn't break the balance, because you could do everything from this list anyway. But the things you're promising to do would be clear from the start, so there won't be a feeling that you're just playing dumb with your opponents.
Also, this would give you a new challenge :)