Will you have a car? If so, staying in/near the old town is more of a challenge because you have to park the car somewhere. You can't drive inside the old town itself - you would have to park outside the walls and drag your bags in.
You can park close to one of the gates just outside the old town (which isn't that big) - that's probably what I'd prefer to do myself. Read the reviews of various accommodations on Trip Advisor that will give you an idea of how far it is from the old town, how many steps (if any) issues parking a car, etc.
FYI, most people who stay near the old town in Dubrovnik don't stay in a proper "hotel" - they stay in sobe (private rooms), apartments, or B&Bs. There are very few proper "hotels" right in the old town - more further away. But I prefer the non-hotel options myself: usually cheaper, more friendly, etc. Sometimes hotels are a good deal.
On my first trip to Dubrovnik I didn't have a car and stayed at a soba just outside Pile Gate - no steps, great location. On a recent trip, I had a car and stayed in the nearby suburb of Lapad and parked my car at a hotel there (great bargain price) and took one of the frequent buses into the old town, just so I didn't have to hassle with a car.
I highly doubt everything is booked for April; more likely, many of the places you are checking either haven't opened up bookings for next April yet or aren't open for the season until May. April is a pretty slow time on the Dalmatian Coast. Tourist season starts in June. I was there in May and things were just picking up. I wouldn't panic about booking something early. I booked most of my accommodations less than 24 hours in advance - very easy to find places (I drive through Croatia and Bosnia over a few weeks).