How will weather conditions affect our cross country move from PA to OR?

No way of knowing whether you'll have weather problems until a few days beforehand. I wouldn't plan a longer route unless you wait until the last minute and base it on the forecast. No matter how far south you go, you could encounter bad weather at that time of year over the High Plains and the Rockies. On the more northern route, the potential for bad weather extends out further eastward onto the plains, but the chances are you'll be OK.

I have driven the western end of i80 many times in spring. I advise proper winter gear, extra water, warm sleeping bag, a shovel and some food in case you get stuck in some spring blizzard. Those can be pretty intense with heavy wet snow and lots of slushy and icy roads along with whiteout conditons. This winter has been really dry and warm; and is expected to stay that way. Also i80 is a big transportation corridor. Most likely if you hit bad weather you'll be stuck in a hotel for an extra night, these storms pass quickly.

any advice as to how to cope with such a big move would help me It's hard to say what would help without knowing what is scaring you / stressing you out. Of course, there are lots of sources of stress in a big move, so it's possible that there are several things stressing you out or you're not sure what is stressing you out. I'd say the first thing is to accept that it's okay to feel stressed as long as you don't let it stop you or control you! I'll address stress from the move itself: the best way to reduce stress from a move is to build in a few days or even a week of flex time. Don't try to start work the day after you plan to arrive in town. If it were me I'd build in a week or two to get used to my new area, but if that seems excessive then give yourself a least a few days. If you are shipping stuff from PA to OR, make sure that your plans can handle you being a day or two late or your stuff being a day or two late. As humanfront mentioned, you could be stuck in a hotel for a blizzard for an extra day. That probably means planning to arrive a few days before your stuff does and having some essentials like sleeping bags or an air mattress. Even better, if you build in lots of flex time you can take some time to do sightseeing as you travel. So my recommendation would be to plan on taking a nice long time regardless of the route you take (maybe build in an extra 2-3 days for sightseeing), then build in a few days to get settled before all your stuff arrives, then build in a week or more to get your stuff unpacked. As for the route, you might be able to change the chance that you'll hit snow by taking a different route, but percentages don't mean much - whether or not you hit snow matters. And you could hit snow on nearly any route that makes sense for PA to OR. So I'd take whichever route you want to, whether it's the shortest or the one with the most sights to see, and just plan on stopping somewhere an extra day to let the weather pass if the weather gets bad.

Once you've crossed into Oregon, generally the section of I84 you need to worry about is that between Baker City and Pendleton. That's crossing the Blues. It can get yucky during the winter, but honestly, it's late enough in the season that it's not too likely to happen. It's not absolutely impossible, but it's really not likely. I drove that section this weekend and it was warm enough that I had the AC on. Weather is unpredictable and it could be awful when you come through, but so far this has been an incredibly mild winter (much too mild, given how low the snowpack is). There's no option better than checking the forecast a few days before you leave and picking the route based on that. Both routes you are talking about are major trucking routes, so they never shut down for very long. Bring money for a couple extra hotel nights and you should be fine.

Also, take a look at the answers in http://ask.metafilter.com/276475/Best-least-treacherous-driving-route-over-the-Rockies.

Thanks everyone! Also thanks Penguinicity, I hadn't found that post in my previous searching for whatever reason.