Well from my experience in the sleepers, the best time to shower is between 2 and 4 in the afternoon. But still in the morning other people will want to at least wash their faces, brush their teeth, and comb their hair. That makes the hours between say 7:30 and 9:30 busy.
Also the same is true at night as people head off to sleep. So those hours would not be good times to spend a long time in the bathroom, working on a sponge bath. Yes, you'll want to do the other things that I mentioned above too and that's ok. Just don't add even more time by adding the sponge bath at those times.
While I certainly wouldn't suggest that you deliberately wake up at 5:00 AM, if you do find yourself up at that hour or anytime before 7:00 AM, then by all means take advantage of using the bathrooms.
As for upgrading to a sleeper, there is no easy way to answer that question. It would depend on how far you traveled in the sleeper. If you just spent one night while on the way to Chicago, you might be able to get a sleeper for $100 to $150 extra. If you wanted both nights, expect it to be more.
If you ask the conductor nicely, he might be willing to give you a few different prices for different lengths of your journey. Amtrak prices the rooms by the night, miles traveled, and what meals are included. So waiting until after dinner to move from coach to a sleeper could affect the price. However, you might find that staying in the sleeper the next morning until just after lunch, only adds 10 more bucks to the price.
I do suggest however if you want to try this, that you do it on the first part of your trip to Chicago. The train from Chicago to Boston is far more likely to have all of it's bedrooms already sold.
For that matter, while it is extra paperwork for him/her, a really nice conductor might even be willing to sell you a bedroom for just part of the day from say right after breakfast till just after dinner. While you of course would not get to sleep in the sleeper, it might only cost you 40 or 50 bucks for that short period of time.
That would still include at least lunch and dinner for free in the diner and you would have time for a shower. I can't however promise that you will find a conductor who wants to do that. It is extra work for them.
But hopefully you'll meet a conductor who is helpful and into making his passengers comfortable. Tell him that you don't have a lot of money, but you'd like to see if you couldn't get one night in a sleeper. If that cost isn't too high, then ask him how much more a second night would be. See what combinations he can work out for you.