Is there a specific type of Medicaid strictly for long term care?

Yes, it's called 'long-term care Medicaid'; Google that phrase for New York and you can see the application and the eligibility requirements.

https://www.google.com/search?biw=1920&bih=937&sxsrf=ALeKk021OJhWCmemjSYK2dLKA83LobHWTQ%3A1608934506880&ei=amTmX9apNfaF9PwPvcCMiAk&q=long-term+care+eligibility+assessment+new+york&oq=long-term+care+eligibility+assessment+new+york&gs_lcp=CgZwc3ktYWIQAzIICCEQFhAdEB4yCAghEBYQHRAeMggIIRAWEB0QHjIICCEQFhAdEB4yCAghEBYQHRAeMggIIRAWEB0QHjIICCEQFhAdEB46BAgAEEc6CQgAEMkDEBYQHjoGCAAQFhAeOgQIIRAKUP-XBFjKpwRguboEaABwAngAgAGKAYgBogiSAQMwLjmYAQCgAQGqAQdnd3Mtd2l6yAEIwAEB&sclient=psy-ab&ved=0ahUKEwjWhv-NlOrtAhX2Ap0JHT0gA5EQ4dUDCA0&uact=5