"I became sick today."
If you use the x不x question pattern, the verb can’t be modified by an adverb. So you have to say either 也行吗 – or 是不是也行, but I think the version with 吗 sounds a little better here.
According to my native Chinese consultant, 我今天... and 今天我 ... are both correct, or at least in use. In any case, if thee is an order rule, it seems to be inconsistently applied. ...