Can I bring my stepson and stepdaughter from the Dominican Republic to live in the USA if I’m a US citizen?

The minimal requirement for filing petition for your step children as a US citizen is that your marriage to the father of the children must have taken place prior to the 18th birthday of the children such as would qualify them as children of US citizens. You have not stated what your husband's view is on this issue and it would seem that particularly because you are now separated his views on this matter would be crucial. They are his children and if they enter the US, they are also his responsibility - something he might or might not be prepared to undertake.

Yes. As long as you married your stepchildren's father before their 18th birthday and are still legally married to the father, you can file a petition for them.