You can only get SSI if the total amount you are awarded for SSDI, is below the full benefit amount for SSI ($674). I assume this is your case? If your unemployment and SSDI put you over the maximum benefit amount for SSI, you can't get SSI. When your unemployment ceases, and if SSDI pays you less than the maximum for SSI, SSA will pay you the difference in SSI benefits.
Be sure to apply for SSI benefits at least 3 months before your unemployment ends, so they will have time to process your application. Be sure to inform them of the date your unemployment benefits end. You can apply after the benefits end, but you will be without that additional income you get, until the SSI starts, plus the retroactive pay from when unemployment ends and SSI is approved.
PS While waiting the 2 years for Medicare, if you are uninsured, most states have Primary or Physicians Care services. You get basic medical care at the county Health Dept Clinic, plus a pharmacy discount card. Some doctors will accept PC cards for treatment. Check with your local Social Services and/or Health Department to apply.