What is the ruling on not performing ghusl when experiencing misgivings about the emission of semen, accompanied by feelings of fear and anxiety, and is prayer acceptable in this state, and what are the ways to get rid of this anxiety and prevent other misgivings?

You have done well by turning away from obsessive thoughts and not paying attention to them, because dwelling on them is a cause of evil. The cure for obsessive thoughts is to ignore them. The anxiety you feel is unnecessary because you are doing what Allah has commanded. This anxiety is a deception from Satan, and you should be reassured that your prayers and acts of worship are accepted. You must cut off the root of these obsessive thoughts. Sheikh Al-Islam Ibn Taymiyyah advised seeking refuge [in Allah] and remaining steadfast in the truth, for Satan is repelled when he sees the strength and steadfastness of the servant. You are not required to perform ghusl unless you are absolutely certain that semen necessitating ghusl has been discharged. With doubt, it is not obligatory.