The best solution is as you suggest to put in some drainage. Make a ditch across the lowest point which leads into a large soakaway if there is no obvious point at which water can run away.
You can make drainage paths leading to the ditch with a mole plough. This is a 3ft long blade with a cylindrical 'torpedo' on the lower end which is dragged through the soil to create a drainage hole and a slit through which the water can escape. It may need a tractor to do this.
Alternatively you can dig channels leading to the ditch and fill these with gravel.
This is quite a radical solution but from the sound of it your land is permanently waterlogged.
An alternative is to build tall raised beds above the ground as these could be made to drain easily and the moisture at the base would be useful for keeping plants ok during droughts.

If the garden slopes towards your house then the developer should have put in adequate drainage as it will affect your houses foundations over time. Check with them that adequate drainage was put in - sounds like it is not!

Agree with Kildermorie that you should check back with the builders - I don't know if you have a similar system over there, but here, there's something called an NHBC guarantee, which means any problems in the first year with a new build, the builder is obliged to sort them out by law.