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Very few people can successfully make 3 full and proper speed sessions work per week. One speed session is not enough. The fact that you already understand that your body is not able to handle 3 to 4 is a good thing. Means your output is enough to have a tough time repeating. Remember to find alternative ways to achieve similar goals.
Going out hard? When I hear that language it makes me think you might be doing something opposite to what you need to do. The essence of speed and sprinting is not ever hard. All you are trying to do is execute what you are doing in practice ( which sounds like it’s going well btw). And picking it up… RB made the point that smooth is very important , staying relaxed and if you perceive gear changing there is too much thinking going on. I am really not so sure you are " doing" anything wrong. Sometimes things just take time and all the pieces come together. I am not so sure this advice I am giving you is overly helpful but when I hear someone " trying " to do a bunch of things instead of just going out and repeating their training I think it’s reasonable to mention it.