So I know the Oblivious Turing Machines head position depends on the size of the input word and a number of steps. Can it be modified in such a way that it's not dependent on the size of the input word?
According to the WP article on Turing machine equivalents, oblivious machine head position is independent of input and is a function of time instead. To wit:
An oblivious Turing machine is a Turing machine where movement of the various heads are [sic] fixed functions of time, independent of the input. In other words, there is a predetermined sequence in which the various tapes are scanned, advanced, and written to.