I notice that I tend to pronounce the /t/ in Japanese laminally (at least I believe this is the appropriate term), especially when it follows /ɕ/ as in "した". Whatever it is, it certainly feels different from the /t/ in English, and I am wondering what the proper way to articulate /t/ and /d/ is.
It is. The WP article of Japanese phonology says:
/t d n/ are laminal denti-alveolar (that is, the blade of the tongue contacts the back of the upper teeth and the front part of the alveolar ridge)
If I may add something, the position of Japanese t, d, n normally varies from back of teeth to the border of teeth and alveoli. The completely alveolar (i.e. doesn't touch teeth) articulation of Standard English would sound unnecessarily heavy to Japanese ears.
A good way to simulate Japanese //t// sound is to put your tongue at th's (//θ//) position, then shift your tongue a little bit forward so that it touches upper teeth.
Though //t// sounds in many languages are often casually called "alveolar", the official IPA chart makes no difference between dental, alveolar, and post-alveolar stops, because few languages distinguish them internally (you can put diacritics if needed). English alveolar /t/ is relatively backward even compared to other West European languages.