My dictionary (Большой универсальный словарь русского языка) indicates that дождь can be pronounced in two ways /doʂtʲ/ as it is written or /doɕ:/ as if it were written дощ.
Does that hold true for oblique cases as well? In other words can дождя be pronounced /dɐ'ɕ:æ/ as if it were written дощя or perhaps with voicing /dɐ'ʑ:æ/?
Shortly speaking - no, even if someone pronounces /doɕ:/ и "по дождём" most likely "ж" will be clearly articulated - there are some variations though where /d/ could be omitted or "ж" will be geminated and/or palatalized.
However, as it was mentioned in this post, the pronunciation norm in Russian is not set in stone and in some cases there can be surprising variations. If I would hear something like /dɐ'ɕ:æ/ I would most likely recognize it though might find it odd.
Two-consonant [ʐdʲ] ([ʂtʲ] unvoiced in Nominative): Зимовье зверей - Ход дождём - multiple occurences in nearly all grammar cases. As the singer is from Sankt-Peterburg (then Leningrad), it could be accepted as excellent example of this norm.
Similarly: Игорь Корнелюк - Дожди - rhymed "подожди" and "дожди"; may be also treated as SPb variant. (But it could have also rhymed with [ʑː] in both, with the same success.)
A likely Moscow variant with [ʑː] is noted in comments to the question.
Recorded dialects include variants as [ʐdʲʑ], [ʑdʲʑ], [ʑː], [ʐː] as well (with respective devoicing).
Anyway, the main thing here is that this consonant sequence is normally voiced (and can be voiceless only at word end position).
Fully voiceless implementation is, to compare with, in Ukrainian (and respectively spelled: дощ - доща - дощу - ...); common norm requires [ʂtʂ] here ([ɕtʲɕ] in дощі).
