What should the questioner do regarding her prayer and purification given the difficult circumstances she is experiencing?

One must hasten to pray on time. Wudu (ablution) should be performed normally and is not affected by the impurity of the place. As for the prayer itself, one must seek out a pure place. If no such place is found and the ground is impure, then if it is possible to purify the spot for prayer, do so. If it is not possible, then pray according to your situation: if the impurity is dry, you should bow and prostrate; if the impurity is wet, you should motion for bowing and prostrating. However, if the dirt is not an impurity, such as mud or dust, then you pray normally, bowing and prostrating.