Check out RC preschools for affordable rates, also local park district classes that would allow you to register for 4/6/8-week classes (no commitment to ongoing fee payment). The YMCA offers preschool programs and preschool. Check if you will be anywhere near one. Your twins are young though and you will be at home. Three is a more usual age to start a few half days a week of preschool in your circumstances, and four much more likely (again, often for just a few half days a week).
You may also find classes offered at museums and your local library will be a major resource for you. Look at facilities like ice rinks for classes too, and tennis clubs. Check out the location of local parks.
I would take the apartment they are offering too, and you can scout around during the six months for something more suitable when you get the pay of the land.
When it comes to venturing out with twins and a baby, please don't hesitate to look for recommendations for teenage mothers' helpers, and be prepared to pay the going rate. Teens under age 16 are your best bet for this sort of casual employment as they are not eligible to get payroll jobs either FT or PT except with exemptions, but after 16 they are and many don't have time for babysitting any more. Many kids aged 13ish to 16 do babysitting and mothers' helper jobs. You should prioritise finding someone you can call on for a few hours a week.
There has been a massive outbreak of measles in Washington state www.npr.org/sections/health-shots/2019/02/08/692665531/measles-cases-mount-in-pacific-northwest-outbreak and a lot of concern about anti vaxers, so I would expect a climate of raised eyebrows for you if your DCs are not up to date with an American vaccination schedule. You will typically register your DCs with a pediatrician for routine care in the US, though there are Family Practitioners too (roughly equivalent to GPs).
If you have your baby in the US a hospital pediatrician will supervise care of the baby in the hospital. Sometimes this hospital pediatrician will have openings in their practice and you could sign the DCs as patients. Or the hospital pediatrician might recommend a colleague. When choosing a pediatrician, look for an office that has experienced and knowledgeable nurses who do excellent phone triage - you don't want an office that says 'Oh bring little Johnny in and we'll take a look' when one of the DCs has a high temp, only to be told it's a virus, you go home along with every germ in the area and get really sick, and then the bill for your visit arrives... Better to speak on the phone to a good nurse who is paid to weed out virus cases.