Sayadaw U Tejaniya's approach emphasizes a continuous, relaxed, and wise awareness throughout all daily activities, not just during formal sitting. The method you described—watching feeling continuously, acknowledging thought, putting it aside, and coming back to the feeling—aligns with his core teaching of maintaining continuous awareness and acknowledging the thinking mind without judgment.

Key elements of his teaching relevant to your description include:

• Continuous Awareness: The practice is about maintaining awareness all day long, not just on the cushion. He stresses that continuity is vital for meditative mind work, not intense, forced concentration. You are constantly reminding yourself to be aware.
• Acknowledging Thoughts: He explicitly states, "Don't feel disturbed by the thinking mind. You are not practicing to prevent thinking, but rather to recognize and acknowledge thinking whenever it arises." The mind that is thinking is just another object of awareness. By acknowledging it, you don't get lost in it. The focus is on the observing mind that is recognizing the thought, not the content of the thought itself.
• The Importance of Attitude: Sayadaw U Tejaniya emphasizes checking the attitude of the observing mind. When you are watching the feeling and acknowledging a thought, you should check if the mind is doing this with a relaxed attitude, free of tension, greed, or aversion. Trying to force the thought away or cling to the feeling would be an example of a wrong attitude.
• Learning from Objects (Feelings/Thoughts): By continuously observing the feeling and acknowledging the thought, you are gathering information. You are learning about the nature of both feelings and the defilements that arise with thoughts (like desire or aversion). You watch how the thoughts affect the feelings and vice versa, leading to wisdom (dhamma vicaya—investigation).
• Returning to the Object: Your description of "coming back to the feeling" after acknowledging a thought reflects the perseverance required to keep the attention inward and continuous. When awareness is lost, you simply go back again and again.

In essence, the practice is a relaxed, continuous observation of the mind-body process, where all experiences, including feelings and thoughts, are objects to be acknowledged and learned from with wisdom and a non-reactive attitude.

Here is a relevant video: Am I Thinking or Observing the Thinking? This video directly addresses the part of your question related to recognizing and observing the thinking mind in Sayadaw U Tejaniya's practice.