Joe's suggestion is very good, but sometimes it's not practical to remove the subfloor or it may actually cause other problems. Either method is good.
Years ago homes were built with 1/2" cheapo ply subfloor then covered with a good quality 5/8" underlayment to receive the vinyl floor once the construction mess was over. In the era your house was built most builders changed to a single and thicker subfloor system and let the flooring company prep for the vinyl.
A single 3/4" might be stiffer than two 1/2", but you'd be fine with either. If you just add 1/2" B/C exposure 1 ply, yes of course you have to offset the seams both ways. Do not glue unless you can do a 100% lamination, (very doubtful), and space the sheets about 1/8" Normally underlayment is not fastened to the joists, only to the subfloor, but with only 1/2" subfloor the screws may not bite enough. You won't know until you you do it.
Absolutely no paper or plastic between sheets. Is there a basement, or crawl? Follow installation directions for any moistures tests for the bamboo.
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