I never stayed home with my son even when potty training. Real life experience is what kids need, not insulation. Yes, it can get messy but whose life doesn't. You still carry a diaper bag but put plastic bags, a couple of changes of clothes (include socks and an extra pair of shoes cause pee runs down the legs and pools in the perfect little resevoirs the shoes make). When you get somewhere, scope out the landscape. Find out where the bathrooms are. My son is 9 and can still tell you the fastest routes to any public restroom in town because of his bladder control issues that are beyond his control. Once you know where the bathrooms are start there. Before you leave, go to the bathroom. When you get somewhere, go to the bathroom first. When you are away from home carry a timer you can set for 1 hour. Most kids can hold it that long. make the timer the bad guy. When the bell rings, you hit the nearest bathroom to make an attempt. To show him you mean business you may have to try as well in the beginning. In our family, EVERYONE went when the bell went off. My daughter (much older) hated it but it really helped my son to see us try as well. Sometimes I would grumble like he did "I don't have to go but the silly bell went off." Then after I would say "Wow, it's a good thing that silly bell went off, I had to go after all". Remember when you are in public all parents have had children in that stage. Not one child throughout time has gone from diaper to complete potty training without any accidents. If anything happens, puddle on the floor or whatever, just notify a store clerk so they can notify housekeeping and maintain store safety, apologize, say something like "sorry, he's potty training, where's the restroom?" Then go clean up, pack your wets in plastic bags prepacked and go finish your shopping. Yes, I have had to buy shorts and underwear while at the mall cause I forgot to bring any with us but such is life. I personally did not have time to sequester us at home while going through this process. You can even give him a star on a chart for each trip with no accidents and after 10 trips he can have a toy or something. At 33 months, he should be old enough to understand the concept of working for a delayed reward. If you have not introduced this to him, this is a prime time to do so. Some people think this is the equivalent of bribery but our jobs do this every time we get paid. We work for two weeks before we get our check. Not many people get paid each day they are on the job so begin preparing him for real life now one task at a time.
Good luck,
C.