Are you sure its not audio getting from output back to input while its in record, and therefore adding an echo to the signal? You could prove this by recording at the front of a reel, and then at the back. Print through is the adjacent layer of tape printing, so at the front of the reel the print will be short delay, and at the end it will be long. If it doesn't vary, then its feedthrough, possibly in the in/out assembly. Use a burst of tone, easy to hear the starts and ends then.
Does the in/out assembly have its transformers fitted? Any funny loading on the outputs? Missing or dodgy ground to the in/out assembly?