Start by triggering on behaviour rather than on time alone. Somebody who requested a quote, visited a pricing page, or asked a specific question has told you something, and a follow up referencing that is a continuation of a conversation. A message sent because seven days elapsed is a message about your calendar rather than their situation, and it reads that way.
Make the first message the most useful one rather than saving value for later. The common structure sends a polite check in and reserves the substance for a subsequent message, which means the recipient decides whether to keep reading based on the weakest thing you sent. Lead with the answer to the question they were probably about to ask.
Give them a route to slow it down that is not unsubscribing entirely. A line offering to stop, or asking whether the timing is wrong and when to return, converts somebody who would have disengaged into somebody who tells you to contact them in March. That information is worth more than the sequence, and almost nobody asks for it.
Stop when the sequence ends rather than restarting it. The behaviour that generates genuine irritation is indefinite pursuit, and it is usually the result of automation with no exit. Decide the number of messages in advance, and when it concludes, move the person to a list that receives your ordinary material rather than continuing to pursue.
Then check what it produced by reading replies rather than by reading rates. Open rates measure whether the subject line worked. What tells you whether the sequence was welcome is whether people respond, and what they say when they do. A sequence with respectable metrics and no conversation is one people are tolerating rather than reading.
Exclude people who have already replied, which sounds obvious and is the most common failure in small business sequences. Somebody who answered your first message and is now in a conversation with you should not continue receiving the automated follow ups, and receiving them signals clearly that nobody is paying attention. Check that the exclusion works by testing it yourself rather than assuming the setting does what it says.