Some stations have a lot of lockers and if you are going to be at a hotel or ryokan for several nights you can also avail yourself of takyuubin. I personally like to shop in Japan (we collect Japanese fashion dolls) so I use it often. I know many people here like to do everything with just carry ons, but I expect they are not bringing home the same things as I do and I also only like doing laundry on vacation so much.
In terms of visiting some sights on the way, I went from Tokyo to Hiraizumi and then onwards to Morioka to meet a friend for dinner and I hit up most of the sights in Hiraizumi without a problem. I didn't make it to the gorge, but I was also starting from Tokyo not Sendai which makes a huge difference.
It takes about 100 minutes or less to get from Oishida to Yamadera. So you're looking at 160-190 minutes total. Most ryokan kick you out by 10:00 am, so even under worst case situations, you can reach there by 13:00 which should give you a good 3 hours to see everything.
BTW do you know about Hyperdia? It's a great site for figuring out trains. http://www.hyperdia.com/
I've not personally been to Ginzan, it doesn't really appeal to me, but that's OK, this trip is about what you want to do, not me. I have been to Zao, in '09 during GW. I loved Zao, the famous rotemboro there is fabulous, however, it's not really a "cute" traditional town and it is very vertical. I think I also visited Shinzaemon no Yu at Zao, along with the onsen in my low end ryokan and I rented a private bath as well. Tohoku has a lot of onsen. It really depends on what exactly you are looking for. Supposedly, they also have a famous skunk cabbage area in Zao, but I didn't make it, just to the one near Tazawako
But based on people posting here about Ginzan, I know it fills up because there are only so many facilities.
Here are some possible itinerary ideas:
Mon 27 April - Ginzan Onsen (Day 1 of my JR East Pass) (night Ginzan onsen or substitute Zao)
Tues 28 April - Travel to Sendai and visit Yamadera on the way (pay out of pocket it's 1860 yen for the entire JR portion Oishida to Yamadera to Sendai) I don't know what the bus to and from Ginzan costs but it's not covered by any pass. From Zao it would be even less and a shorter trip.
Wed 29 April - Sendai - Matsushima Bay (pay out of pocket it's 410 yen each way from JR Sendai to Matsushima Kaigan Station)
Thurs 30 April - Sendai - Hiraizumi on the way to a hotel near Tazawako (It's only 30 minutes longer to Tazawako and you may as well stay there since everything you want to do is near there) (Day 2 of Pass) There are hotels near Tazawako, I was looking into it last year, but stayed in Hanamaki 2 nights instead.
Fri 1 May - Tazawako to Kakunodate (1420 yen one way on the shinkansen and 15 minutes)
Sat 2 May - Take bus and overnight in Nyuto onsen and see the skunk cabbage first if there is time (night Nyuto onsen)
Sun 3 May - Long day Nyuto onsen back to Tazawako to Hachinohoe to bus to Lake Towadako (overnight there) and Day 3 of Pass
Mon 4 May - Work your way back to Tokyo, you could stop at Nikko on the way (day 4)
Tues 5 May - You can use the 5th day of pass to get from Nikko back to Tokyo or use it the day you go to Kakunodate.
Another option would be Sun May 3 go to Aomori and overnight there (day 3 of Pass)
On the 4th go to Oirase and either use the pass or pay out of pocket for the bus, but go round trip back to Aomori and overnight there again or take a late train back to Tokyo or Nikko, but make sure you can arrive late where ever you go or spend another night in Aomori.
In terms of GW travel, if you are traveling that day just book the ticket as soon as you get to Tokyo. In general, it is easier to book Tohoku trains than Tokkaido line trains. Any Japanese who are traveling to the north are more likely to take a plane from Aomori to Haneda because it is much cheaper than the 17,740 yen it would cost by train. So if you pick a train and book it the day you arrive in Japan or the next you should be fine unless you are a huge group.
You can get Oirase in, but its much further from Tazawako then you think and the fastest way is possibly rental car because some bus routes are just not that common. If you are staying overnight I would stay in Aomori over Hachinohoe because Hachinohoe is sort of small and quiet, unless that is what you want. When I went to Oirase a decade ago, I stayed in Aomori, though I admit, if I go back again, I will probably drive so I could hit up Oyu onsen again.
A lot of the regional passes are great deals as long as you are just staying in that one region. Just decide which days make the most sense to use your 5 days.