There are two possible Halachic reasons why we should wash our hands after touching shoes.
1) There is a mystical reason. An evil spirit rests on shoes. This is connected with the verse Gen. 3:17 "...cursed is the ground because of you...". Therefore, shoes which are the focal point for touching the ground (and feet) causes a bad spirit to rest on the hands that touch them. (see Kaf HaChaim 4: 60; 70-71; Shulchan Aruch HaRav; Rav Ovadiah Yosef)
2) Shoes (and feet; some say sweaty dirty socks too) usually touch the ground and become dirty from inside with sweat and outside with dirt of all kind. Therefore, touching them requires handwashing by Halachah due to an obligation of physical cleanliness. There is no mystical issue. (see Shaarei Teshuvah and Mishneh Berurah on 4:18.)
The Poskim bring both opinions and many are lenient to assume it is only due to cleanliness. One thing seems clear though. The shoes are a problem even if they are someone else's and even if they are not currently being worn(Pischei Teshuvos 4:22). A partial hand touch seems to be enough to cause a problem too (Kaf HaChaim 4:64).
One thing is clear though. There is only an absolute obligation to wash if you are going to pray or learn Torah or be involved with some other davar sheh'b'kedushah. (of course, sticking your hand into that bag of chips wouldn't be advisable either. :) )
True, some advocate that we are still worried about the mystical reason. So, according to them, there is a benefit in washing right away, (to quickly remove the evil spirit) even if you don't plan to daven etc. until much later. But if not for the mystical reason, then all washing in Shulchan Aruch is for cleanliness before kedusha, like prayer.
But, if the octo-mom is busy multi-tasking or running a daycare, it seems she can rely on the opinions that only require her to wash when needed for something like davening or eating.
There are also the following leniencies proposed by many Poskim: (But, there are those who argue to be strict anyway.)
a) New shoes (or very clean shoes; if we don't worry about evil spirit)) should be OK. (No evil spirit from the ground, no dirt either) So trying on shoes in a shoe store is not going to cause a handwashing. (Sdei Chemed, Chalitzah:2)
b) Shoes that were never once worn outside have not touched the ground. Therefore there is no evil spirit. However, cleanliness could still be an issue. (see Aruch HaShulchan 4:21 Pischei Teshuvos 4:22)
c) Shoes that are not made of leather are exempt. (see Aruch HaShulchan 4:21)(The Steipler and Rav Shlomo Zalman Auerbach discuss if this is accepted or not.)
d) Shoe laces and the tops of boots are exempt. (Pischei Teshuvos 4:22 note 215; on boots) Chazon Ish, Rav Shlomo Zalman Aurbach; Halichos Shlomo Tefilah, 20:18, Rav Ovadiah Yosef; Yabia Omer 5:1 about laces.
e) Clean baby feet should be exempt. Your own clean feet should be OK (if you ignore the evil spirit issue too)
f) You only need to wash the hand that touched the shoe. (see Kaf HaChaim 4:86)
g) You only need to wash once with water. (not 3 times if you are not worried about the evil spirit) This can be accomplished by putting the hand under the faucet (no negel vasser cup needed). (Rav Ovadiah Yosef)
h) If you have no water, you can rub your hands off on something that has friction to remove dirt. Yeshivah bochurim are always rubbing the wood of their shtenders if their hands get ritually dirty instead of running to the sink, because they don't want to fall into the sin of bitul Torah. (losing time from study) (see Shulchan Aruch 4:22)
i) Touching via gloves or a rag is ok. (see Kaf HaChaim 4:87)
Finally, we might invoke the merit of Count Valentine Pototski. He is also known as Avraham ben Avraham. He was a Polish noble who refused to give up his Judaism and was burned at the stake by the Church in 1749 (or Avraham ben Yitzchak in 1753??). Rav Shlomo Zalman Auerbach has a tradition from the Vilna Gaon that after his death, the world changed. Through his merit, a Jew no longer needs to fill negel vasser by the bed, but instead may walk to the sink in the morning; because the power of the evil spirit has been reduced globally in his merit.
Just a thought. If that's true about one Ger Tzedek, then what about the holocaust? Would the mystics pasken that the evil spirit has been removed so that hand washing about the evil spirit on shoes (and other things) is no longer an issue either? I have not heard.
I hope this helps. :)