Selling Dimes/Quarters In Volume

I agree with specksynder.

My local coin shop was buying 90% common silver coins at 25X face today.
So your 300 coins would bring you $1125.00 at that shop.

High grade will be worth more than melt usually, but unfortunately people dont believe in premiums moving in sync with silver prices. So you are likely not to get much more than melt.

The ANA show just starting is absolutly not the place to try to sell those. And you would have to pay $6 to get in, aabout $11 to park and then find out most dealers there really want to sell, not buy unless dirt cheap. Dealers at that show pay from $500 to over $2,000 per table so they really have to sell to make up for that. Also, they usually stay at a hotel, buy food, etc. while there so that too must come from selling, not buying. Most dealers there are out for big profits due to all their expenditures so buying low grade coins is just not in their interest. Usually.
This is why almost all coins at that show are overpriced. For selling your bulk coins, try some of the local coin shows just outside Chicago in the suburbs. One last Sunday in Countryside would have been the place for your coins. On the 21st, there is a coin show in Itasca, straight up Irving Park Road where many local coin store owners are present so your coins could sell there.
You mentioned you live in Chicago. Aren't you aware of the constant 3 coin shows a Month in the suburbs. Also, a massive amount of coin stores.

I have this feeling that if silver prices remain high for long enough that it is possible that so many common date coins could get melted down that some of the dates may become rare enough to start gaining numismatic value.

As for even getting melt value, that can be tough too. Refineries aren't going to pay full melt value for the coins (probably 90%), meaning that anyone that wants to buy the coins for melt is going to pay enough below melt that they will make a profit even after the refinery takes their cut. The only way I see getting full melt value is if someone wants to speculate either on the price of silver continuing to go up, or take a chance that some key or semi-key dates might be in that bag. Dealers are unlikely to do either.

Obviously you do not know how many Silver and/or Copper coins are melted every day. Silver coins are constantly melted by jewlers to make Silver Jewlery. They really normally have no idea of a coins value except what they can make with it and sell for jewlery. I've seen Jewlers melting coins right out in the open. Not hidden at all.