No, in general as the nominal bike size goes up the head tube and top tube also get longer, meaning the handlebars are higher and a bit further away. For the style of riding you are doing that isn't terribly critical and if you are comfortable on the bike for the length of rides you're doing then there is no reason to get a different bike.

Most people find that they want a new bike after riding regularly for a year or so, because in that amount of time you figure out exactly what you do and don't like about a bike and the odds of the bike you started with being just what you end up wanting are pretty small (regardless of what that first bike was). So it makes sense to wait before thinking about another bike for that reason anyway.

If your Roadmaster has a relatively flat top tube then you may very well be right about the current seatpost being long enough. I was thinking of modern mountain bike geometry like this:



That bike had a 400mm (15.7 inch) seatpost to accommodate the height shown. I definitely still think it is worth marking your insertion point and pulling it out and measuring to see how deep it is. If it isn't in far enough you can break things, including yourself.

Regarding your comments above about not feeling nimble enough for jumping on and pedaling from a stop, I think you'll discover pretty soon that it's not actually a problem. When you get on the bike and start at the beginning of the day you aren't starting in the saddle. Getting going from a traffic stop is basically the same thing. It may feel awkward now, but in a couple of weeks it will become second nature and you won't even think about it.