Houston Airport: 1 hour transfer sufficient (terminal C to E)

Hi, I got a question: planning a trip and the proposed connection is a domestic flight arriving in terminal C with 1 hour to hop on the inernational flight departing terminal E.

• Same airline, so I assume I don't have to pick up and re-check-in luggage, or do I?

• Travelling with wife and 2 little kids (5&7) – is it realistic to transfer in 1 hour (late afternoon)?