From what I've seen wallet addresses have started with 1, then 3, now they are bc1q ... I guess I'm wondering do I need to shift the funds to new paper wallets? Or, keep them as is and hope a future bitcoin wallet can still accept them?
Not really. I don't think there will be a time where legacy wallet addresses(with
1's) will be sort of deprecated and sort of "disabled", knowing that hodling is a widely adopted strategy. If there was plans from the devs and communities though(though I heavily heavily doubt it), I'm pretty sure it won't be activated over night, and there will definitely be a lot of arguments about it that you'd have a good amount of time to move your funds.