More on the bike making a difference - here's my quick history of commuter bikes. I just started bike commuting in Spring 2013. You'll see I end up where some others in this thread have suggested - on an older steel road bike.
1. Old department store full-suspension mountain bike with knobbies that had hung on my garage ceiling for years. Heavy and slow. I had to get off and push it up one hill on my first day. I wasn't all that fat, but quite out of shape from sitting behind a desk and driving to work for many years.
2. I found this forum.
3. Bought a '90s rigid mountain bike from craigslist. It came with knobbies. The lack of suspension and lighter weight made it feel very fast, indeed.
4. Put the knobbies on the shelf for future actual "mountain biking" and installed smooth street tires on the rigid mountain bike. The bike was MUCH faster on the street and hard-packed dirt trails. I couldn't imagine riding a faster bike. It FLEW...or so I thought.
And then...
5. A good friend who is coincidentally the same size as me got a new road bike and gave me his '90s steel road bike. I took it to a LBS for a full tuneup, new chain and cassette, bearings repacked, etc. Eventually I'll learn to do that stuff myself, but I had full-on bike fever by this point and wanted it done quickly and correctly, and I was also delighted to learn how relatively inexpensive bike maintenace is compared to car maintenance.
The road bike is pretty light and goes up hills practically effortlessly, compared to the rigid MTB with street tires. At first I thought I'd get out of shape again riding such a light bike that just rolls forever with so little effort, but of course that hasn't happened. I ride it more and for longer distances. It's wonderful, and I can't recommend a road bike highly enough.
Sorry for the long and self-indulgent post, but I was where you were about a year and a half ago and a light and fast steel road bike turned me into a out-of-control BIKE FANATIC. Why didn't I know cycling was so much fun before?
If I were you, I'd take the old Schwinn into a good LBS and have them do a complete overhaul. Keep the receipt. If you don't like it, put it on craigslist as "LIKE NEW CLASSIC ROAD BIKE! - PROFESSIONALLY REBUILT" or something and use the proceeds to help fund something you like even more.