Before start answer, I want to clarify that I am not an with mantras or such rituals. I do not practice regularly such rituals nor with the mantras. So I cannot give an authoritative answer.
However, this stotra is in sanskrit and I am acquainted with that.
tam = To that
namami = I bow down
shanaishcaram = slow mover
So it translates to "I bow down to that slow mover(Lord Shani)." This makes sense grammatically because 'tam' has a accusative form. It means preposition 'to' is part of the form. Here, 'tam' becomes a predicate in the sentence. And 'namami' implies subject to be 'aham'(means 'I' in english), even though it is not explicitly mentioned.
tvam = you
So it will translate to "I bow down you slow mover(Lord Shani)." Here, 'tvam' has nominative form. 'Tvam' would become the subject and hence makes it grammatically incorrect since 'tvam' is second person and 'namami' is in first person. Hence grammatically, 'tvam' should be the word.
shanim = to (Lord) Shani
pranamyaham = pranami + aham (by sandhi rules)
pranami = I bow down
aham = I
it translates to 'I bow down to that Lord Shani'. Grammatically, this it is sound. here 'aham' is the subject in first person. 'Pranami' verb is also in first person. Hence they match. Similarly, 'tam' and 'shanim' also match as they are in accusative form.
The entire stotra is describing Lord Shani and has descriptive clauses. This may have been a poetic gesture. And hence it justifies demonstrative form, 'tam'.
Comment sections have suggestion 'tam namami shanaishcharam' which is grammatically correct. But, the question has only 2 options
tvam namami shanaishcharam
tam shanim pranamyaham
I would choose the second option based on Sanskrit grammar.