Yes, worry can become sin. The first flicker of anxiety usually isn’t, though.
When Jesus says “don’t be anxious about your life” in Matthew 6, I read it as an invitation to trust. Sin at its core is turning away from God. Worry becomes that when I choose to keep rehearsing my fears instead of bringing them to Him.
I also don’t think “all sins are the same” holds up. Scripture treats all sin as separating us from God, but it doesn’t treat every sin as equally serious in its effects. An involuntary spiral about your future isn’t in the same category as cultivated hatred.
What’s helped me is Philippians 4:6-7, taken literally as a practice. When the spiral starts, I stop and name the specific worry in prayer, with thanks for something concrete.