Then one day, the mother of the woman called her daughter to check up on her, but didn’t get an answer; so she decided to come by and see how her daughter and grandbaby were doing and to her dismay she was disappointed and hurt by the situation that her children were in.
The semicolon seems out of place. I think you could replace it quite nicely with a comma. I think if you leave it there, then you really should follow the so with a comma. More formally it would read, "... didn't get an answer; therefore, she decided.." and the semicolon looks right in that instance. Since so and therefore are used the same way, perhaps we only need to add a comma after so to make it correct.
Grandbaby is not a word. Use grandchild.
The sentence is getting pretty long. There are longer sentences in literature, but it's perhaps more readable, and urgent-sounding, to break it up. In this case, I would break it at this point, changing "doing and to her dismay..." to "doing. To her dismay...
I forgot her name, but she was nice person and all, but tends to make some bad decisions; such as buying drugs instead of using that income on something far more important; her child.
Generally, you can only use one but in a sentence without confusing me. Delete the first one here, add a second she to the other one and it makes good sense to me: "I forgot her name, she was a nice person and all, but she tends to make...."
The first semicolon is inappropriate here; it should be used only to tie two complete, self-sufficient sentences together more closely than a period/full stop would allow. In this case, after decisions, either delete it entirely or change to a comma.
The second semicolon is wrong for the same reason: "her child" is not a complete sentence. Use a colon instead: "something far more important: her child."
The child was skinny as if he had a case of malnutrition; and the house was a mess as clothes, trash, and drug residue were dispersed everywhere.
You do love those semicolons, don't you. It's not necessarily wrong here, but odd. Much better to use a full stop/period. Either way, you should drop the and. If you like the and, then change the semicolon to a comma.
Although you have properly used the plural were after the list of plural items, by concluding that list with singular noun (residue), when you could just as easily have concluded it with a plural noun (clothes) missed the chance to write a smother phrase. There is nothing wrong, only awkward, with your use of "drug residue were" - it would sound better as "clothes were."
Although the mother of the child tried her best to defend her case and keep custody her child, the judge decided that it was in the child’s and her mother’s best interest that the grandmother to be granted full custody of the child; while the mother goes to rehab.
Generally we speak of one's best interests (plural). Perhaps this is simply a matter of taste.
And there is that semicolon again! Since "while the mother goes to rehab" is not a complete sentence, it cannot carry its weight alone on one side of a semicolon.
There are two principal uses for a semicolon. The one you are trying to use here is to take two complete, independent, well constructed sentences and tie them together more closely than a full stop / period would allow. Test it by replacing it with a period. If it doesn't work, then a semicolon won't work either. A comma probably will.
The second use is in a list of things that have commas in them already: men, women and children; alligators; and fish. I am speaking (rather facetiously perhaps) of three entities: men, women and children is one. Alligators is another. And fish is the third. I want to list them, but there is a requisite comma between men and women. To show that group is a single entity, I can't use a comma between children and alligators; I need something stronger. And so, here comes the semicolon to the rescue. But once used in a list like this, you must continue to use it for the balance of the list; hence, it appears again between alligators and fish.