My passport is still valid for another couple of years, but I only have one empty page left on it for stamps. Do you think I will have any trouble traveling with it?
You definitely need ONE blank page, so you could probably get away with it, but if there's time before you leave maybe you can get extra pages added? It's hard to say if you would need them, partly it depends on your nationality and where you are travelling from.
Getting a new passport outside of your country usually takes a week or two longer and this could be a great disruption to your travel plans. If you have more time at hand, I'd strongly recommend getting a new passport before you leave. The visa alone covers one full page and you'd need another for the stamp upon arrival.
Yes, you should look into renewing your passport. Sometimes, you can buy additional pages. I actually ran out of room while in South America, but it did not seem to bother Chile, Uruguay, or Argentina. They just flipped through, saw there was no room, and double stamped on a box. Then, it got a bit wild and started in the margins. Thankfully, I was able to get to the embassy and arrange a new passport. Which by the way, looks awful. The new US passports look and feel like they are made of cardboard, and I hate how they put the information one page into the book, rather than the first page. Now, I have to always flip through to get to the details. But I was pressing my luck. Go get a new passport.
Might as well renew your passport right now. You would rather not have that looming over your head while exploring through the different locales. A quick pit stop, if you will, just so you could set your affairs straightened out. It may be a pain now, but you will be thankful that you did it when you are looking back at it.
I guess you need to renew your passport, just to be on the safe side. Or perhaps you can have more pages attached to it. If you have time, then have it processed now so that you won't have to worry about it in case you have to travel again.