I have field bindweed growing in my landscape, Asian Jasmine, Indian Hawthorne, Loriope, Nandina, Burford Holly, Mary Nelle Holly, Carresa Holly, Cleara, and mixed flowers. I have very minimal spots of it in the hybrid bermuda lawn. I have tried to use round up 1.5%, which I can use in the Asian Jasmine, but I can not seem to get control in areas with other plants. Pulling does not seem to slow it down much. I also have what I believe is purple spurge, flat on the ground fine reddish stems with small dull green leaves. Seems to like it dry and open areas.
Bindweed is indeed a noxious weed and very hard to control as you have experienced. First, in the bermuda lawn you should be able to kill it using MSMA. This herbicide will eliminate most everything in the bermuda lawn without hurting the bermuda. You may experience some discoloration of the grass but it will quickly recover. In the other areas where you cannot spray it with a glyphosate herbicide such as Roundup you could consider treating the individual plants by painting or sponging a 2 to 2 1/2 percent mixture of Roundup on some of the leaves. Pulling it isn't going to do any good as it will reestablish itself from the roots.
The plant that you describe as spurge is probably just that. There are quite a few varieties but if it bleeds white sap when you break off a piece of it, it is surely one of them. You should be able to control it with just about any broadleaf herbicide or with glyphosate. It is an annual that can also be controlled by the timely application of a pre-emergent herbicide that is labeled for broadleafs.