Hi Shadowmichael, I have never done a whole blankie of this type, but have done an 8" or bigger square of it as a square to be used for an afghan where multiple squares were to be sewn together. So I understand the concept.
I had restless leg syndrome going on tonight so I got on the computer to calm it down before going back to bed. LOL So I’ve been doing a little swatch of the “dish cloth” blankie idea. I don’t know if you want intarsia if you are thinking stripes. Stripes are not intarsia. But your mention of intarsia was interesting and I tried that out too. 
Usually intarsia is done in stockinette stitch, but it works for the garter too.
If you want stripes you can change back and forth making narrow stripes of two rows quite easily. That way the yarn you need to change to will always be right where you need it and carrying it up the side only over 2 rows seems to produce a hardly noticeable carry. So I think that would be a good idea. You can make several rows of one color to start out with if you’d like and whenever you want to start striping just start working with the new color leaving a tail, and you don’t really do anything else special to add in the new color. From then on just switch to the alternate color everytime you get back to where it is hanging. That will make 2 rows of each color and a nice effect. If you make wider stripes you would have to carry the yarn along the side and it would show up more obviously and not look as nice if you did, say, 4 rows in a stripe, or 6. So If you would like wide stripes, I would cut the last yarn color at each place and begin knitting with the new one, as above, but you would have more ends to work in.
There is a way to knit in tails as you go. You hold the tail to be worked in up along side the left knitting needle and knit under it when you knit the first stitch, then over as you knit the second stitch and keep alternating like that to catch it in. Do that for about 3 inches and then pull on the tail to neaten it up. It shows a bit, but may be satisfactory.
If you change every two rows you can avoid the need for working in a lot of ends and it will look nice.
Intarsia is usually used to make pictures or blocks of color surrounded by a background color. When you have seen horses and buffalo knitted on the backs of sweaters, that is intarsia. I introduced a little block of intarsia on my test swatch of the garter and it works. But with the blankie getting bigger all the time unless you made the intarsia block get bigger all the time it would end up making a stripe running diagonally across the blankie, or you could make little squares or whatever all over for a possible interesting effect and fun to do. (Or I suppose you could do a horse, or buffalo. LOL) To do that you could use short sections of the “block” color and you have to start a separate ball of the background color on each side of the “block”. I don’t think that is what you had in mind.
When you change color in garter, the change shows on one side. So there will be a right side and a back side, but the back won’t look bad, just different. The color changes will show instead of being such a sharp color change as there will be on the right side.
Have fun with your baby blankie.
Usually intarsia is done in stockinette stitch, but it works for the garter too.