Can I use my approved visa for this new flight considering that I'm
still going to be in Dubai for less than 96 hours and still flying
you obtained a transit visa means you are transiting in UAE and your final destination is Seattle, if you simply want to use this visa from India-UAE-India means it will no longer be a transit visa.
From official Embassy OF United Arab Emirates :
"The Transit Visa could be granted to a passenger transiting to a third destination through one of the UAE airports by the concerned authorities of the airport. This type of visa will allow the passenger to stay in the United Arab Emirates for 96 hours or less."
Further another online resource Emirates 247.com confirms the same for UAE transit visa policy that itinerary to and from the same country will not be applicable:
According to an email from Amer Contact Center, a passenger can apply for a transit 96 hours visa at the airport upon paying its fees. However, they have to meet certain conditions such as the duration between the two trips should not be less than 8 hours and the two trips should be to different destinations (one should transit from Dubai to another destination and not back to the same place where he/she came from.
Lastly I have confirmed from a reputable travel agent from my home country and he says the same, therefore I am afraid you cannot use this transit visa simply for Mumbai-Dubai-Mumbai.