Titebond III, gorilla glue, or anything similar should work fine. Just takes a long time to dry so your parts will need clamped or pinned and always checked for being straight and square. A small craft glue gun and some glue sticks from Wal-Mart could end up being the same price.
If you have a Hobby Lobby, Michael's, or Joann Fabrics near you, I'll recommend that you try there for a glue gun. Why? Because they constantly have coupons online for 40%-60% off of one regular price item (and the glue guns are almost NEVER on sale). You can pick up a quality glue gun for $15-$20 using one of those coupons, or cheaper if you want something that's not regularly priced at $25. Just make sure you get a full size glue gun; the mini glue guns just don't cut the mustard for the amount of glue you'll need for most planes.
Which reminds me...I need to hit up one of the aforementioned stores for a Maker Faire I'm working next weekend, building FT chuck gliders for kids; I need more glue sticks.
I recently finished my first build, I can't fly it yet, but I am pretty good at crashing. I don't offer much experience at all. but I built the simple scout with white gorilla glue. I did use small clamps and books and jigs to hold it in place. But through all my abuse, no joint has failed nor weakened or torn. My power pod moves around a bit, but that is skewer issues. It will slow you down but since my time was limited I would glue up late at night for several nights.
The other issue with using some glues is that they eat foam - that is, they dissolve it. For example, using super glue? Unless it's a foam safe formula, you can pretty much count on it dissolving into a gooey, plastic mess.
Is there a reason you don't want to use hot glue? I mean, you certainly can use a different glue, but it's just not as effective or is more time consuming, that's all. Just curious if it's due to electricity issues (you're building at a location that doesn't have readily available AC power outlets), or safety issues, or cost?
I build without Hot glue because of many reasons including the long time durability of the joints and the somewhat extreme climates that my country experiences.
Initially I tried the UHU-Por but it was way way too expensive for the quantities I would be using and so I sought a lower cost equivalent and found a class of glues that work very well with relatively low cost.
It works on Foam Board, a wide variety of the foams found in current use for RC Model aircraft as well as with lighter and porous woods like Balsa.
If you are able to seal the FB surfaces prior to assembly and gluing then you can use cheap packing tape to hold the pieces firmly in alignment so that you can glue large numbers of joints and set them aside until the glue hardens/dries/sets. You can be frugal with the glue application and a bottle will last many models and as stated previously the weight of the glue in the model is quite minimal.
As for the glue joint strength I find the glue binds well to the foam and generally in a crash the FB rips and fails whilst the glued joints remain intact. It is so great for foam that I use it for repair of crashed foam, (retail), models and the repairs can be near invisible and incredibly strong.
Should you finish your models with a coat of paint over the sealant, (minwax), then the FB itself will stiffen as well and you will end up with a quite long lived model, (if you do not crash or otherwise damage it!. My son has a number of my creations using the method and materials I mentioned above and he stores them in a metal shed/garage. The temperatures inside exceed 60 degrees C in mid summer and yet the planes are as good as the day they were place there and fly the same as the day they were built.
I prefer the hot glue only because of how quickly it takes to hold. In some cases, when you're doing a large wing build (like a Kraken or the Sea Duck wing) having to press and hold for however long it takes for the glue to dry can be a pain, especially if it's glue that takes 30 minutes or more to fully dry. This means you have to find weights that are heavy enough to appropriately hold it in place but not heavy enough to crush the wing, or you may be holding it down yourself for that time...
It's not unheard of to use regular wood glue, as mentioned above, just be aware of some of the pitfalls of working with it.
Also be warned that gorrilla glue does not stick to the waxy coating on the waterproof foam board. It pops right off with your fingers. Does white/wood glue work with it? I've been meaning to make some test pieces for that.