2 Rountrip flight from Houston to Bangkok and Phuket to Bangkok

Hello! My fiancé and I are planning a trip to Thailand and I was wondering if someone could help me figure out some flight questions I have. We will be traveling from Houston and plan to stay in Bangkok for a few days then fly down to Phuket for a week, then go back home to Houston. It’s expensive to book a multi-city flight from Houston > Bangkok>Phuket>Houston. So, I was thinking of booking 2 round trip tickets: 1. Houston to Bangkok 2. Bangkok to Phuket. The idea is to fly from Phuket to Bangkok and catch a return flight from Bangkok to Houston.

If we book the 2 roundtrip flights, will we need to pick up our luggage and go through security when we fly into Bangkok from Phuket before we can catch the return flight to Houston? Does that make sense? Please, let me know if I need to clarify anything!

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