So I'm not an expert, but I did take immunology last semester... You're definitely right that we'd benefit from more study of the long term downstream effects of adjuvants. When you put the immune system on high alert, it sometimes remembers the wrong thing from the experience and continues to target it later.
But you've got to weigh those risks against the risk of dragging our feet on rolling out new vaccines. It's likely that many autoimmune diseases are caused by pathogens which are either undiscovered, or which are known, but whose link to the autoimmune disease is undiscovered. This is because there is evolutionary pressure encouraging pathogens to resemble self-antigens. If a pathogen has markers that cause it to resemble the human liver, the human adaptive immune system won't typically attack it because it wants to avoid attacking its own liver. So the pathogen is safe. This is called molecular mimicry. But the mimicry isn't perfect, so sometimes the immune system does become sensitized to the pathogen, attacks it, clears it, and starts slowly attacking the liver (or whichever tissue bears the same epitope as the mimic). It shows up years later as an autoimmune disease, but the data just isn't there to isolate its cause to that pathogen.
So the best strategy, if we can pull it off, is to vaccinate against the original pathogen (or prevent it via other means... clean food and water, etc), but to target epitopes which do not resemble any human tissues, and with adjuvants that do not sensitize the immune system against any human tissues.
As for how long adjuvants stay in the body, it's definitely worth studying more, but their whole function is to sound the alarm, so I'd expect that they're cleared rather quickly. Anything that fails to do that is by definition not an adjuvant. But assuming that an inappropriate response was triggered in the first place, you'd see the same effects whether or not the adjuvant was still around: If the initial exposure has triggered clonal replication of whichever B and T cells attack the antigen, self or otherwise, they'll stick around in your lymph nodes for years and years, and they'll re-up their numbers with each subsequent exposure.