Insurance and speech therapy is a tricky situation, and I'm not sure with Medicaid.
You can refer the boys to your local school district at any time. There should be a preschool team in place to assess them and provide therapy - it depends on your district what they tend to offer. The assessment would be free and take place at the school, and then you would have a meeting (not the same day, they would schedule it) with everyone. You can bring anyone you want, and they will bring the school team members, including at least one teacher and the speech pathologist.
If they get therapy through school it will be with an Individual Education Plan (IEP) which outlines their goals and services. Most schools don't offer full-time school programs to kids unless they have multiple areas of delay or extremely severe delays. Some like to offer preschool with less therapy, some tend towards appointment therapy and preschool only for kids with more significant needs. This could mean a two or three hour program three days a week, or group speech therapy you bring them to for half an hour two times a week - it all depends on their needs and your district. You are free to disagree with anything they offer, accept it, or accept part of it (like accepting speech and turning down preschool) depending on what you feel comfortable with. You are also legally allowed to observe the classroom they would be in prior to placement if you want to.
As far as at home strategies - I would try a lot of focusing on a few words at a time. So in the middle of an activity choose a word or two and use it to death, in fairly short sentences. It helps understanding, and imitation or use. So like "Blocks go up. Go blocks! My blocks up! Crash blocks. Oh no blocks!" I'm not saying to use this kind of speech all day, just for a short period to focus on words you want to help them with. It's easier to imitate one word at a time, and also easier to learn a word you hear in context, with lots of intense stimulation.
To help with putting two words together, you could try modeling two words, especially when giving them choices. If they repeat back one word, just say the two again.
"Drink milk or eat apple?'" "apple" "Oh, eat apple!"
You can also expand what they are saying. So after your ds uses one word, repeat it but add to it just a little. If he says car, you can say 'go car' or 'my car' or whatever makes sense at the time. This helps them figure out the whole sentence thing, especially since it's just a step above where they are, and close enough to try. They might not imitate it at first, but just keep modeling!
It sounds like a tough situation, it's so hard to watch our children struggle. Hopefully you can find the help for them you are looking for.