There are essentially two different viewpoints under which you can look at the extra dimensions:
Full-blown string theory, in which we treat all dimensions as equal and stuff propagates in all directions it can without distinction.
Effective field theory, in which we perform compactification/dimensional reduction to obtain an effective field theory in fewer dimensions from the original 10-dimensional string theory. "Propagating" through the extra dimensions is not a meaningful notion in such an effective theory.
It is crucial to note that full-blown string theory is not a quantum field theory and the Polyakov action is not an action for a QFT. You should not think about the worldsheet coordinate fields as the "same kind" of fields as the fields in QFT. While the quantization of the string and the rest of string theory take a lot of "inspiration" from QFT - unsurprising given that it has to reduce to QFT in some limit if it is to describe our universe - they are not a QFT, first and foremost because the worldsheet coordinate fields are functions on the worldsheet, not functions on spacetime like in an ordinary QFT. The proper QFT fields on spacetime you obtain in the low-energy limit of string theory are very different from the worldsheet coordinate fields, in that they are the fields of a ten-dimensional supergravity theory rather than those of a two-dimensional conformal field theory like the worldsheet fields.
Depending on the nature of the compactification, in the passage to the lower-dimensional effective theory we have another process that can produce different fields that are physically relevant. The "particles" we see in our world are the particles associated to these effective fields, not to the 10d SUGRA fields or the worldsheet coordinates. It is perfectly meaningful to say that the particles associated to one of the 10d SUGRA fields propagates through one of the extra dimensions, avoiding the paradox the question poses.
Finally, note that the worldsheet coordinate fields are precisely that, coordinates - each of them represents the position of the string in a certain dimension, not the whole dimension, so I do not agree that "a dimension propagating through itself" is a proper interpretation for anything here.
