Should I get a BS in IT and then a MS in Computer Science or a BS in CS and then a MS

I already have a BA in an unrelated field, but want to change careers to be a database programmer and eventually a Database Administrator (DBA).
I can’t even apply for a MS in CS unless I have a BS in either IT or CS.
A BS in CS requires 31 classes including several classes unrelated to computer programming including Math and Science classes.
A BS in IT requires 13 classes total with only a few programming classes (2 are in Java), but I would be learning computer programming on the side anyways as I am already learning SQL and will learn other languages like Python, etc. on my own as well.

I will be working full time while going to school and studying for computer certifications.

I am asking because I am 33 years old and I would be close to 40 if I go with the BS in CS + MS in CS as opposed to the BS in IT + the MS in CS, not to mention the expenses, etc.

I suppose it depends where you live and where you want to work.

All of the web development shops I want to work for don’t require any type of certification, as long as I can do the work and demonstrate the personal / technical skills for the job then I qualify.

It may be different for you but I would certainly not choose to study six more years if I were 33 years old for a change of career.
If you’re really serious about doing this then you should be able to learn all the required skills in 1 year of dedicated study, maybe 6 months.

I’ve always thought of schools as a place to teach you how to learn, if you’ve already done this then there’s only the practical skills that you can still learn from it - and those can be done faster and cheaper on your own.