Carbs left sitting with gas in 'em for 7 months, unfortunately probably need a cleaning of the pilot jets, the left one in the jet block. The orifice is tiny. It easily becomes plugged, and idle-low speed operation is affected. While you might think &*%$$@! small pilot jets are plugged-up is an evil way to extract $$ from your pocketbook, it's a fact of life in the use (or, perhaps non-use) of a VMax. I'd have to say that learning how-to tear-down a quartet of VMax carbs to the two-pair point, so you can remove the float bowls, and the jet block, for a thorough cleaning, is about as-important as learning that as you pass 6,000 rpm it's time for an E-ticket ride, and you don't want to be on sandy pavement, wet pavement, oil/ATF-covered pavement, you get the idea. I suggest buying four jet block gaskets before you open things up, and the needle valves for the floats. OEM is best here, though I've used the K&L needle valves with no problem, they're available separately, while you have to buy the OEM needle valves and seats as a set, which will cost you much-more. The guys on-here who rebuild carburetors as a business will say, "OEM parts only" and you're likely gonna have no issues going OEM. Read another member's issues with an aftermarket set of carburetor rebuild components, he sent 'em back.
Yes, you likely have a bad 2nd gear engagement dogs set. The male dogs are worn, the female holes into-which they engage are worn, and the easiest thing to do is to avoid use of 2nd gear, shift 1-to-3: skip 2nd gear. Engine removal, and splitting the cases is the only way to proceed here. Consider getting the gears undercut, it helps shifting, and while it costs $$$, while the transmission is apart is the time to do it. You might want to look for Damon's media on eBay motors. He has 'em for cleaning/rebuilding carburetors and yes, the engine, to-fix the 2nd gear 'no-work no-mo' problem.
I forget where you are, please consider adding your location below your screen name. Someone close-by may have a resource available to you.
Did you replace the gas filter? You can buy one for ~$7 from your local automotive parts shop, a generic sealed cartridge filter, or you can get one off ebay that is rebuildable, with a sintered bronze replaceable element. Russell I think had the design, there are ebay motors knock-offs, do a search on the forum, or just search eBay. If you get the throwaway, open your to-be-discarded cartridge, to examine what it's been catching.
Just from a point of information, shine a light into the gas tank, and what do you see on the sides/bottom? It should be shiny-clean. If a prior owner tried sealing it, and did a crappy job, you won't see that. There are various acid/alkaline ways to clean gas tanks, electrolysis, and mechanical methods like media blasting. Search on the forum for more info. If you have a rusty gas tank, it's useless to clean the carburetors, without fixing the gas tank, or replacing it, if it's got perforation.
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