To share how bad it can be, sometimes I used to worry about that verse so I was worrying about worrying!
Sometimes if I prayed about something that at the time was worrying me and if it came back into my mind, I would pray about it again and the cycle continued. Do you know that Satan is the great accuser? My spiritual attacks mostly seemed to focus on this: 'the bible says you have not to worry and pray and here you are worrying, you are a rubbish christian.' So I would feel defeated and then feel worse. (when the thoughts came as 'you' I knew 100% it was a spiritual attack)
Then I realised what was happening and any kind of attack like this I just say this prayer and look in the mirror* "I am saved by the grace of God, not by any works I can do or will do as it will never be enough to earn my way into heaven. Thank you God for this! I am a work in progress and will never be perfect this side of heaven but yet you love me and keep working in me. My weakness actually shows God working through me as I give over something to God - others around me will witness what I find difficult but then see God working in me." (or just quote Romans 8:1!) Since I have started doing that, those thoughts have not been coming that much!
The past few days looking and mediating on this verse, I saw it a different way. That it wasn't an admonishment about worrying but more of a 'you're not alone in all this, you can come to me and I will help you.' I knew this in my head but not in my heart or practically as I always felt guilty if I was anxious as people usually quote that verse to you and then I felt worse as I had prayed about it.
If you are anything like me maybe you will understand and my goal in sharing this is to help you.
I think sometimes that God just wants us in continual prayer with Him and while in some things I have instant long lasting peace when I pray, other times I have peace but different aspects come to mind that I am anxious about so I come and pray about it again. Those times God just knows better than me how much I need long continual prayer - alone time with Him and I think it's just one way of doing it, letting me know that I need that special closeness with Him at that difficult time. Do you need to know this as well?
So you will understand why I say quite frequently that it is a great daily example to me, watching dogs living in the moment. So that in those really tough times I actively am reminded to "Be still and know that I am God" and live in that moment giving it all over to God.
*Sometimes when you are having such negative thoughts I find it helpful to look in the mirror when reinforcing the truth as well as praying. I look in the mirror rarely re this but it may help you at those really dark moments in your life.