Can I Obtain A New Visa Without Leaving Russia?

No, not legally. Here is the explanation why.

An entry visa to Russia is processed by the Russian consulates in foreign countries. No entity in Russia can issue a Russian visa.

Sometimes people think it a good idea to send their passport from Russia to their country via FedEx and have the consulate or a visa expediting agency send back the new visa once it is ready. Indeed, this would be the worst possible option to take because in such a case your new visa will be invalid as you will not have crossed the Russian border with this visa and obtained a stamp in your passport.

What with one visa expired and the other, invalid, a foreigner founds himself in a most difficult situation. The only thing he can do is to obtain an exit visa (which is a really complicated procedure) and leave the country directly.