It appears to me that for a newbee (like me) that the simplest and least expensive closure system will be crown capping with a capper and oxygen absorbing caps. Crown caps are also good for long term storage if the bottles are stored in an area of low humidity...to keep the caps from rusting on the outside. Corks can be a little more complex to work with but completely doable if enough effort and study is included. Capped bottles can be stored upright whereas corked bottles should be stood up for a few dates after corking but then laid on their sides for storage (to keep the cork moist). I'm not sure about swing-top bottles...I've heard mention that they're not that great for long term storage but do fine over the short haul.
As fb mentioned, if the carboys are full you really don't have to be in a rush to bottle...mead makers age in bulk to produce meads with more consistent quality/taste rather than individual bottles that can have different qualities and tastes.
3+ years? I'd say it's ready to bottle.
I'm not sure what to tell you about stabilizing or whether to stabilize a mead that old...I'm a newbee, too.