I'm pretty sure we docked at 9am and made our way down to the gangway around 9:15am. We figured since we had 1.5 days we'd let the "rush" crowd go first. But, they were still there!
If there is a long line at the Bermuda Tourist Info Center to buy your travel passes; I would highly recommend you walk around to the other one! It is right near the ferry docks and I know it's hard to picture by words, but I'll try
When you first walk off the ship you'll walk through gates. As you exit, look to your left and that is where the first TI Center is. If there is no crowd you can buy your passes here, but if there is a crowd, walk to your right, the road will curve and as you round the bend you'll see another building by the ferry docks and your left hand side. The Bus Stop, which will take you to Horseshoe Bay, in addition to other destinations is right before this building, maybe 20 or 30 feet. If it is crowded there will be a long line for it!
With Hamilton having street closures, I would stay away from the area if you're relying on buses, and not ferries, to get you there or back and trying to see more than just Hamilton.
My DH would not "stand his ground" in the line I put him in, which was at the actual bus stop while I went to buy the passes, so by the time I turned around and walked back to him there was a "new" line of people who were pushing to get on the Horseshoe Bay bus!

It did seem as if all 3,100 passengers from Explorer were all in this small area at the same time! I know they weren't, but with the crowds we just wanted to get away from as much of it as possible, so we changed course to something that was less crowded.
If it is so crowded that you don't want to waste time waiting for the bus to Horseshoe Bay, the Maritime Museum sits off diagonal to your right from the Ferry area/bus stop and you could spend an hour there and come back to the bus stop after the crowd has died down. There is a time table for the buses on the bus link I posted above and we found that the buses ran accurate within a few minutes.
Since it sounds like you are going to Horseshoe Bay Beach, it is a long walk down to the actual beach from the Bus Stop, definitely not a problem on the way down as it's all down hill, the way back up is all up hill though! There are people who will shuttle you back up to the top of the hill, these people are parked in the parking lot and have "shuttle to the top" signs. I think they charge $1.00 per person, we gave the driver $5 for the 2 of us because we felt it was worth every penny not to walk back up that hill

Make sure you take some cash with you for this if you don't walk to hike back up that hill. They will wait until their vehicle is full, unless you are pressed for time and offer to give them more. We allowed 30 minutes to get from the parking lot to the top and walk over to the Bus Stop, which has an enclosure and will be on your right hand side once you're at the top of the hill on the main road. We waited about 10 - 15 minutes for the bus, but again the shuttle drivers will want a full van load to take you to the top (I don't know how many are in your party?)
I didn't make it to the other beach you mention or the lighthouse; but we did pass the bus stop you'd need for the lighthouse and we could see the lighthouse from the road. It also looks as if it's a good walk to get to it. I'm mentioning this because I don't know how much stuff you take with you to the Beach? We had 1 backpack DH carried and I had a fanny pack, so we didn't feel weighted down. Some people on the bus had beach bags, sand toys, chairs, etc.