I had 3 intercostal blocks over the last month using a local anesthetic yet I became dizzy after each procedure.?

There are many . Causes of dizziness. The main concern I have is an intravascular injection or rapid absorption of local. It also could be a vasovagal response, related to sedation you may have received, or ortho static hypotension. You should be monitored and given supplemental oxygen if you have another round of injections.
Intercostal Blocks. The intercostal region is highly vascular and it is likely that a amount of local anesthetic was absorbed intravascularly resulting in symtoms of dizziness. Other early symptoms include light-headedness, ringing in ears, ect.