Yes, this has been a tough year to harvest forages as well as to get crops planted in a timely fashion and weeds are definitely thriving. Weed control can be accomplished by either repeated mowing or by use of herbicides.
You did not say what acreage you have or if you took a hay crop off a pasture field. You asked about management of both pasture and hay. In a pasture situation, repeated mowing, 3-4 times in a year can prevent weeds from going to seed and, for perennial weeds, begin to deplete their root reserves so that they drop out of the pasture and allow the pasture grass to become more competitive. Another non-chemical method is to hand pull or hand dig weeds out. This is feasible on small acreages and/or with small weed infestations.
In a hay field, especially for fields managed on a 3x cut/year, weed problems can be minimized by the repeated mowing necessary for harvest. However in some cases, herbicide applications can be helpful to control weeds and allow the pasture or hay field to recover.
With regard to herbicides, you have a couple of potential approaches. You did not mention what acreage you are dealing with, but if it is small enough you could consider spot spraying rather than a broadcast application across the entire field. The weeds you mentioned are all broadleaves and can be controlled with an application of a broadleaf herbicide. These herbicides will kill any clovers or alfalfa in the mix but not harm the grass. One broadleaf herbicide option that would be relatively low cost is a 2,4-D and Dicamba mix. This is also a option for a broadcast spray over the entire field. The advantage of spot spraying is that you target the application and you can be more careful about where you spray and stay further away from plants and produce of your organic neighbor. If you have heavy patches of weeds where not much grass is left growing then using a tank mix of glyphosate with 2,4-D is another option.
Be aware that if removing the weeds leaves a sizeable open area in your field, it is likely other weeds will fill in that hole, so you may have to do a combination of weed control and overseeding after the herbicide application, with grass seed. Always read the herbicide label for any restrictions on how soon after an application any re-seeding can be done.
Some final considerations: When temperatures get above 80 F and especially above 85F herbicides may not be effective because plants are not as actively growing as compared to lower temperatures. Also, taller and more mature plants are often more difficult to control with herbicides as compared to younger, smaller plants.
Please contact me at the Wayne County Extension office at 330-264-8722 if you need more information or explanation.