In a word, no. If you can't spray with round up for woody plants (and you'll need to do it repeatedly) you'll have to get in there and dig out everything you don't want. No other option.

And think about it. Any groundcover that would smother out multiflora rose and euonymous would take over the world--think Kudzu and Crown vetch. The you'd have to get rid of them.

Wow. That is straight talk. How effective can smothering with cardboard be? I have a lot of very large, very thick pieces I was going to lay on top of the cut growth. A waste of time?

I doubt the cardboard would work for multiflora rose or euonymus. But I have cleared a LOT of huge multiflora rose from my property (and done it well into my 60s. ( In my 70s now) I cut back all the branches, hauled them off, then pried out the central stump. Fortunately, that's all it takes--they don't come back from pieces of root. And actually, I prefer clearing out multiflora rose to having to dig out a lot of other invasives.

That is what I am doing, laceyvail, and I am in mid 60s too... Now please tell me, what did you plant in their place and how did you keep them from returning? And must I really bag and trash everything I dig up? Can I pile it up and let it rot as long as I don't use it as compost?

Yes, you can compost everything you dig out, or put it in a burn pile if your area allows that. In my yard, I created gardens in those areas with an enormous range of shrubs and perennials. You can plant anything you want.

An east facing slope with morning sun and shade the rest of the day would want shade tolerant plants--and there are so many gorgeous ones. If deer are a problem, avoid hostas, azaleas and rhodies, but look at Brunnera, Pulmonaria, lamiums, Hellebores, Epimediums, Acteas (formerly called Cimicifugas), and, if there's enough moisture, ferns for starters. There are many, many others as well, perhaps less well known. Shade gardens are often at the their most beautiful in spring. And the foliage is cool and welcoming in summer.

Of course, you will weed some as necessary and until your new plants begin to fill in.