Flying out of IAH to Mexico Thursday at 10AM – Advice on Traffic/Construction?

So my family and I are going to IAH Thursday morning to go on a family trip to Mexico. I have heard horror stories of people getting stuck in traffic trying to get to the terminal due to all the construction. I don't know my terminal yet but I am assuming it will be Terminal E, and we're on United. My app says to check in at Terminal C. We have reserved offsite parking and will be taking a shuttle in to the Airport.

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Anything I need to know? Traveling with two small kids so I don't want to get caught in a cluster. Any recommendations for people that have flown recently, do I just take the Shuttle to Terminal C and pray? Perhaps avoid C,D,E drop off all together, ask to be dropped at A or B and take the underground Tram to C?

Appreciate any advice you all have.