How to verify if a transaction is signed correctly?
Pubkey script alone is not enough to verify a transaction, you'll need:

Pubkey script, to evaluate the scripts
Amount, in order to check if the sum of inputs is bigger than or equal to sum of outputs
Index (in TxOut list) and tx hash of the transaction being spent
Block height so that you can set the consensus rules

After gathering all the above (should be found in the node's UTXO database) you can start evaluating if the provided script is valid which isn't limited to signature verification but running/evaluating the script that includes checking correctness of the script, OP codes, OP count, script size,...
A lot of the verification can be found in validation.cpp file.
This is also how I do it in Bitcoin.Net library.