Best Of

I thiiiiink you might be out of luck when it comes to replacing your phones and keeping your current service plan. Those phones do not have SIM cards, so the plan is kind of tied to the phone. Someone else might be more knowledgeable, but I think that option's kind of gone.

Before I continue with a couple ideas, it's worth noting that if what you're trying to do is a bit out of the ordinary (and it is, data plans are pretty much the standard nowadays), you're probably going to have to come up with something a bit hacky, which probably means a bit of experimentation, or putting up with some quirks.

1. It's been a long time since I did this (think Blackberry era), but I used to do the 'smartphone-with-no-data-plan' thing. My trick was just to get a plan - any plan - that simply didn't have data - so long as it had a SIM card, I could then buy a smartphone whose radio was compatible with the plan (finding this out just involves a bunch of googling, this helps), install the SIM card, and then away I went. This might not be feasible anymore, as 'no data' is going out of vogue.
2. Since the problem is 'without increasing our costs,' you can try looking into a mobile virtual network operator. These are companies that, in a nutshell, sell you a plan but use a bigger carrier's network. Boom Mobile is an example of a virtual carrier that uses the Verizon network, and they have an individual plan that offers no data. Again, experimentation is required - they might suck. Furthermore, it would be wise to use your own phone instead of buying one through them... that way if they suck, it's easier to get away from them. However, using your own phone is costly.
3. You COULD embrace data. No, not as in a 'suck it up and get with the times' way, but by using something similar to Project Fi. The idea here being that your phone number gets ported to Google Hangouts, so that all calls and texts are done over data. Now, I've heard some folks say they use unsupported phones (like a OnePlus device) on Fi and it WORKS, you just end up only using one half of the service (ie, the OnePlus One only has one radio, so it wouldn't be able to benefit from switching back and forth between networks). This is a pretty cheap alternative.
4. In the same vein as number 3, it's possible to port your number to google voice (which, however, may be deprecated in the future) and make all your calls and texts via hangouts. In this situation, all you need is the smallest LTE data plan - no talk, no text (because all that goes through data). This is actually what I'm doing right now.

Each of these idea has its own caveats - you risk running into devices that Simply Don't Work with what you're trying to do, or companies that won't grandfather you in sometime down the line... but experimentation and janky ideas is part of doing something nonstandard.