Puddles indicate poor lawn preparation. In our winter wet climate lawns should have enough contour to drain without puddling. It also indicates your topsoil is too thin and not enough subsoil contouring or tilling. It's a common homeowner mistake. If professionally done, it was done poorly.
Your lawn must be large to justify the cost of having it hydroseeded.

Without a picture or two, or being there, I can't answer your question as to whether or not your soggy spots will dry out.

2 days is not long enough for anything to dry out this time of year. Once the soil gets wet enough, which it has by now, it will not be dry again until June or so.

Grass does not have enough root mass to it to dry out wet spots. If you have marshy spots, and you want something to suck the water out of the soil, you need willow trees or cedars or alders or cottonwoods. Grass won't do it.

I agree, sounds like inadequate soil preparation. Clay does not drain. Topsoil laid on top of clay won't drain either.

What is Terraseeding? I've never heard of that before. Is it different from hydroseeding?

A new trend in landscaping is to plant 'rain gardens' in low spots that stay wet. Not a lawn or a tree, but a group of plants that like wet soil, like salmonberry and douglass spirea and sedges and rushes and Japanese iris. I think the King County website has information on planning and planting rain gardens. It's something to think about doing if you have spots of poor drainage.