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Welcome to Vimarsha! There is a sweetness of devotion and a delight of aesthetic experience found here that is hard to find elsewhere, you're already clearly sensitive to it, and will just find that attraction becoming stronger and stronger. We'll have a new live course this fall and, like all of our courses, they are absolute free with no-strings-attached to attend them live. So budget your time for study and reflection, not your money! It will be on one of the most important and esoteric elements of the tradition, so certainly stay tuned! There is a difference between nāma or "name" mantras (like Namaḥ Śivāya) and bīja mantras that have tantric seed syllables (like hrīṃ). As a general rule across all tantric lineages, the former type is fine for non-initiates, the latter aren't.
The chanting of haṃsa along with the breath is also certainly fine. For more details on this, you can see the commentaries on the Vijñāna Bhairava, which gives this practice in its first several dhāraṇas. Jaideva Singh or Mark Dyczkowski's translations are both excellent. The short of it though, is that you don't have to chant the syllables, they're actually recited by the breath itself, you rather just have to attune yourself to hear them in the sound of the breath coming and going. If you use some ujjāyī breath to start out with, you can hear it more clearly. The breath comes in from the outside into the body, and stopping at the heart center (its specific location is behind the sternum or, more precisely, xiphoid process), and then returns back out starting from that point.
Keep sticking around and asking questions or engaging on the forums with your questions and the fall course will be ready for us all in no time!