From my understanding, segwit blocks send transaction information without the witness, which includes the signature. I thought nodes needed the signature to verify the transactions?
This is not correct.
All segwit blocks do differently is having the witnesses (which include the signature) in a separately-hashed area. They're still an integral part of the block, and full validation cannot proceed without the witness data. Removing or altering the witness data will invalidate a block.
What this does permit is stripping the witness data without affecting validity according to pre-segwit validation rules. This is the basis for using segwit as a blocksize increase (the old 1 MB block size limit thus only applies to the non-witness part, as old nodes don't see the witness data). However, nodes that implement the segwit rules need the full witness.