I am slowly working my tilled plot. today I was hoeing up some weeds in the unplanted area. It is a spreading morning glory.

If I till these under won't they just grow back? I noticed the little white roots while digging holes for tomatoes and sure enough these have sprouted.

Don't ALL weeds grow back after tilling? How else can I deal with weeds in my plot?

If the weed is one that grows from runners then tilling is a problem. Other weeds, that grow from seed will be OK. As long as they don't seed you've broken the life cycle.
You can lay newspaper down, wet it and cover with mulch like straw to suppress weeds amongst your vegies. One thing i do is chip them with a hoe on a hot day and leave them exposed, then a day or two later do it again. This will knock the runner weeds too. Just expoose the roots to the sun on a dry hot day.
Then mulch once they have fully wilted and died and water the bed well. Keep the mulch thick and you'll have minimal weed probs.

is plain straw good or how about alfalfa?

All these weeds I just hoed then I will leave in the sun and in a few days I will hoe again I guess and try the mulching idea.

would that work in an asparagus bed too? I'm preparing that as well. I have some big healthy roots to transplant. I would think the mulch will discourage the asparagus to send up shoots.

Alfalfa is great. Plain straw is fine.
Your asparagus will surprise you. It'll send up shoots thru your mulch, if you lay down on it, it'll push you out of the way ;-) but make it a tad thinner over the crown if you're concerned.
Asparagus likes a well prep'd bed with lots of organic matter. You can place a 'biscuit' of alfalfa in the planting hole to feed the asparagus while it's young. Bury the alfalfa in a bit of compost or good soil then plant the asparagus.

what is alfalfa biscuit? Rabbit pellets or are there bigger biscuits?

I fought wild morning glory for 18 years in my veggie garden at my old house. I used everything I could. I even left the garden bare one year and sprayed it with roundup every week. The morning glory reappeared the next spring - worse than before. I still had great production, it just took constant digging and pulling to keep the morning glory vines from choking everything out.

If I went on vacation for a week, I would come back to a garden that looked like all I planted was morning glory. Then I would have to spend 2 days digging, pulling and unwinding the vines from the tomatos, green beans, brussel sprouts, etc.

Last week I found 2 morning glory sprouts in my mum bed here at my new house - I've hit them with roundup every day since then. They are almost completely dead and I'm still painting the them with roundup every day - just in case. Morning glory is the Japanese Beetle of the plant world.