We're in western Ohio.
Do you have a good stand of grass? The recommendation I'm giving is based on a good stand of grass.
Graze it as needed this year. Spray/clip any weeds that are out of control. Soil test, especially for pH, apply fertilizer as required but skip N. Next spring graze it as close to the ground as possible. In early May pull the cows & drill in your alfalfa. We've used a regular wheat drill with the alfalfa in the small clover seeding boxes. Leave the cows off until the alfalfa is firmly rooted. Graze the grass to get it down to 4 inches. That will probably be the end of June. The alfalfa will outgrow the grass at that point.
We've done that many times with good success. It sounds like you don't have enough alfalfa currently growing to cause an autotoxicity problem with a new seeding. We've used different varieties of alfalfa and now just use the old Vernal. Grazing varieties didn't last any longer than Vernal in pastures for us. Mixed with grass we've never had any bloat problems either.
Another option if you have good soil moisture right now is to soil test, graze close, apply lime/fertilizer(no N) as needed, and get the alfalfa seeded within the next 2 weeks. It can be grazed after it goes dormant, around mid October here. Leave 4 inches stubble through the winter to protect the alfalfa.
Consider splitting the pasture in half and doing part this summer, and the rest next spring. That gives you someplace to put the cows while the alfalfa is getting established.
Some things to keep in mind with this system of seeding, you won't lose production from a well established grass sod while the alfalfa is getting established. The alfalfa will not be the dominate forage, but will provide additional grazing during the summer when cool season grasses slack off. Applying N before and during establishment of the alfalfa will encourage grass growth which competes against the alfalfa growth, you want to give the alfalfa the advantage while it's getting established.
Edited to add: alfalfa grows best in full sun. Don't waste your seed in the shade.