mulch to use for cover crops planting

So, this year I am planting some cover crops mixes in the veggie garden. I have 5 raised beds....4' x 50'. I recently planted a mix of field pea and oats to over winter. However the germination is not so great. I am confident that I got them in deep enough, however I have no mulch over them because I was worried that would hinder germination. Although, now I think that no mulch is almost worse because the sun dries the soil out so bad that nothing is going to germinate unless I'm out there every single night watering the whole garden.(I have drip irrigation but that obviously is no good without mulch). So, any ideas about what mulch and how much I might be able to get away with with out hindering germination? I have the following mulches available: wood chips(fresh and composted), grass clipping from the lawn(fresh and several weeks/months decomposed), and organic wheat straw.

you are going to want a light weight mulch that has enough holes so day time warmth can get in along with some sunlight.
I'd probably use a blend of the ingredients you listed and around a one to two inch depth.
The alternative would be to use floating row covers, which, while satisfactory, is not particularly permaculture oriented, but we do what has to be done to get to the end we desire.

With 1-2" of light mulch do I need to sow more shallow ?

I would not think so. I've seen oats and field pea come up through three inches of leaf litter that fell on them after being planted with a seed drill. Mostly it will be the availability of some light to guide the sprouts up through the mulch, that is very key to using mulches over seed beds.
The light weight allows the new plant to straighten up and come through.