You can easily avoid the Straw Market by just not turning in there. If you look on Google Maps, you can do a search for Nassau Bahamas Cruise Terminal and you will basically see the area. The place I hated the most was the Festival Place since you are forced to go through there. There will be all kinds of people trying to get you to take a carriage ride or see if you need a ride to Atlantis. I just said "No thank you" and continued on. We basically walked Woodes Rodgers Walk and Bay Street with no problems checking out the shops, even checked into some of the small side stores. You can easily take this walk and be back for lunch. I actually walked it twice, once with my mom and once with my dad.
There were many ships in port the day we were there, but it did not feel busy walking around, but I felt perfectly safe. If I find myself in Nassau again and I do not want to do an excursion, I would probably try the zoo or pirate museum if I did not want to be off the ship all day.