When traveling to temples to render services, we maintain Madi to the best of our ability; if by chance we come in contact with someone on the way to the temple, shall we perform mantra snanam (by chanting Apohishta mantras accompanied by sprinkling water on the head) as it is not possible to take a bath at the temple? Is there a sankalpam for mantra snanam? Can we just chant mantram after the invoking the Devata, Rishis, and chandas before sprinkling water?

One has to maintain purity while going for kainkaryam in temples, it is not wrong to come into contact with other fellow kainkaryaparas, but if one happens to touch others it is not possible to take a bath halfway through to temple. Must ensure we do not come into contact with anyone. Mantra snanam is not done through rishi chandas; yes, sankalpam has to be done for mantra snanam.
Notes:
If we need to maintain purity, we should be cautious as well. If the temple is crowded, one cannot avoid coming in contact with others which is accepted.