What is the best way travel in eastern europe right now?

I can understand your hesitations... but rest assured as a tourist (in this case) you will be part of the solution not the problem. I just returned from a few weeks in Poland, and while I did see a few refugees, it was 95% business as usual, there were no shortages, restaurant/shops were bustling. Locals are happy to see you.

As a tourist you are helping to strengthen the local economy, especially in this time of need. Consider many of the seasonal tourism jobs can be a lifeline to many refugees.

Also the situation is rapidly changing, now that the fighting has shifted to the southeast a number of Ukrainians are returning to the Kyiv area. The situation could be dramatically improved by the time you travel.

Here is the link to my recent trip report:
https://community.ricksteves.com/travel-forum/trip-reports/poland-trip-report-april-2022

But I feel awkward about showing up as a tourist in countries that are
now hosting so many refugees.

A1) The refugees will have no impact on your tourism, you wont even know they are there; and the added tourism income will be welcome after COVID and the downturn in tourism due to the war

A2) No one has any idea when the refugee numbers will reduce so dont plan around that, for every one that returns 2 come out and without knowing the plans and tactics of the Russians, you cant begin to guess.

Is there anything new/different than the usual advice on how to avoid
being the "ugly american tourist"?

A3) I know a lot of people in the tourism industry in Eastern Europe and have worked with them for about 20 years now, and it’s not the Americans tourist that they see as ugly; never have in any way or form.

Does the addition of tourist money compensate enough for the added
strain on these countries' overtaxed resources?

A4) Tourism money is the major component of the income for a significant part of Eastern Europe, so you are not taxing the economy, you will be building the economy (in Montenegro for instance tourism is 25% of the GDP); and I suspect you could find very real ways that you could use the trip as a way to help.

Everything I've seen here pretty much seems to indicate 'business as
usual'. Is that really the way it is?

A5) It is and again, I know this from my participation in the tourism market in Hungary and travels to Hungary since the war began and Ukraine, Croatia, Bosnia & Herzegovina, Montenegro, and Albania since the start of COVID, and Romania and Bulgaria and Slovakia just prior to COVID; even the Ukrainians are counting on, and making plans, on tourism to help rebuild when the war is over.

A6) There is no place in Europe I find more interesting and enjoyable to visit (I will be back in June); so if you need help planning, feel free to PM me.

I was in Prague and Budapest last week. business as usual. besides Americans, there is TONS of British and other EU tourists there - so please help Hungary host Ukrainians by giving them your business :P

B1) Shaun Kel; I am glad you enjoyed it and had no transportation issues.
B2) You might have noted, I sort of like the place too; hope I was of some help
B3) I will be back in late June or early July (if i can find a room)
B4) I appreciate your comment about helping the refugees

Thank you, everyone, for your kind replies. It is interesting to hear that some are actually counting on tourism to help.