I'm glad that you aren't replacing the veneer. I do a lot of veneering and it will not work for you because a) the stick-on veneers won't work over an oily (i.e., teak) substrate, b) the substrate is really dirty and won't allow proper adhesion, and c) you won't be able to vacuum bag unless you deconstruct the interior.
I would not paint either. The result is often a job that looks like what it is - an overpaint of what was once a natural wood finish.
Do not oil the wood. That's how it got into the state that it is. It's only dirty and not that bad off. Don't worry about the uneven-ness. That's largely due to dirt. Start by using an abrasive scotch pad soaked in acetone or lacquer thinner to REALLY
clean everything. Then sand lightly
with 150 followed by 220 finish on a 5" random orbit sander or by hand. Do not oversand or you'll go right through the veneer. It's easy to do with the 150. Change sand paper frequently, it will clog easily because teak is naturally oily and the wood is dirty. This is not the time to save money. Finish with MinWax satin urethane on low wear areas and MinWax Spar urethane on high wear areas.
This is not a tough job but will take some time, so do it in sections. Don't tackle the whole interior in one shot, cutting corners because it's getting to be a drag (I hate prep and get bored easily, so I write from experience). PM me with any questions. My interior was in poor shape and with patience and time came out ok. See this link: What kind of interior finish