Wow, that's a lot of melt, Colleen.
You should consider putting in what's called a dry well to drain some of that melt off.
I completed one for a client last year who has had problems with residual melt water.
It was comprised of digging a large hole (I used a small backhoe) about 5x5x5', with two adjoining trenches of the same depth radiating out from the main hole by approximately 20'. Of course, this was for a large area, it can be scaled back to suit the area that's affected.
Vertical 4" ABS pipe with grate covers extend to the bottom of the hole, where they join a "T" that is connected to weeping tile (hose), that extends into the trenches. The main hole, and the trenches, are then filled with clear stone for 12", and then the excavated soil is put back into the trenches, the area is then seeded.
No mechanicals, no pumps, just a means for the surface water to seep back into the ground and then be dissipated into the ground water once more.
Fairly cheap project, assuming that you do it yourself. The main "well", and then adjoining lines need to be below the frost line, which is typically about 4'.
Another means, albeit mechanical, is to sink a sump pump well into the ground at the lowest point in the yard and put a removable grate over the top of it. When you run into that type of a runoff, you can remove the grate and put a submersible utility pump into the sump pump well, attach it to a hose and pump it off down the street to the nearest drain. Using this method requires that you also purchase (with the sump pump well), a watertight lid to prevent water entering the well during the Fall, which will freeze and render the sump pump well inaccessible to the utility pump come Spring.
Both are fairly cheap, and very effective ways of dealing with Spring melt, or with heavy deluges of rain. Both alternatives are a lot cheaper than foundation damage.