Yes, borax tends to raise the pH (and the TA a little bit) which you then push back down using the MA. It's one of the reasons why I don't like the process as people with hard water can experience clouding because of the pH spikes. Once calcium carbonate precipitates into the water, it takes a long time to dissolve (if it ever does at all). Boric acid is a much better product to use for this process as few people care if their pH drops a little bit.
Going from 0 to 50ppm in a medium sized pool (16000 gallons) using only boric acid will only lower the pH by 0.3 units. But, using 1/10th of the necessary amount of borax (so 94 oz) to raise the borate levels will result in an increase in pH of +0.6 and an increase in TA of 11ppm which is why you must add the MA to knock it back down. So, the difference is adding all of the boric acid at once and giving it a good brush then walk away versus having to mix up ten buckets of borax, pour slowly in front of a return and then immediately hit it with acid while brushing in between to make sure everything mixes up really well.
Long story short - boric acid is your friend.....
What is the pH and TA of your fill water? It sounds like you're pool's fill water is the issue. If so, a cover can really make a huge difference in acid demand.