The absolute best solution to this is to convert the PDF to an INDD using PDF2ID from Recosoft. If you get this kind of thing regularly (say, several times a year) then this would be a huge time, money and pain saver. It is quite magical and astoundingly clever at extracting all the information you need from the PDF and converting it to text frames, placed images, and so on.
Because you say you're carving things up in Photoshop, I suspect you're running into Photoshop's fixation on RGB black, which is a built black rather than Registration Black (which is 100/100/100/1000). The various type and black options in ID won't help, because the whole thing is just a placed PSD.
If PDF2ID isn't an option, adjust the file in Photoshop before you bring it into ID. In the CMYK color space, you can use curves or levels adjustments on individual channels to pull down the rich black. (Once the PDF is in Photoshop, unless it was a Photoshop PDF to start with, it's completely rasterized, so you have the individual plates available as channels. That makes it tricky, but you should be able to work with it.) If your text is easily isolated, you can even use curves just on the text, by setting C, M and Y to 0 and K to 100 and masking the adjustment layer so it's only affecting the text.
If you open the Info panel in Photoshop after you've added the adjustment layer(s), you can mouse over different parts of the image and see what the CMYK values are, native and adjusted. Once the image is placed in InDesign, you can use the Separations Preview (
Window > Output > Separations > Ink Limit to see if you still have any over-inking, and you can round trip back and forth until you're below the limit.
If what you have is an actual Photoshop PSD, editable as a native Photoshop file, convert to CMYK and explicitly set the text to 0/0/0/100.
Another possible approach is to use Acrobat. Open the original PDF and convert the entire thing to a black-only profile, such as Dot Gain 15%. Save that as a copy and use pieces from that for your text.