Congratulations, you’ve stumbled across one of the major issues with inch-based tire sizes. Put simply, they’re useless for almost any purpose other than finding an exact match in dimensions to your current tire, because they measure the outer diameter of the inflated tire, which is affected by both the width of the tire and the design of the tire (strictly speaking, they are also useful when figuring out overall geometry of a bike, but that’s not something a majority of people ever need to do, unlike finding replacement tires).
Instead of searching for a 29-inch tire, what you should be looking for is a tire with an ETRTO size of x-622, for any number ‘x’. The first number doesn’t matter much (it’s the width in millimeters of the tire when inflated, so your current tire is a 60-622), the second one is the important one here (it’s the bead-seat diameter of the wheel, also in millimeters, and this is what actually needs to match for the tire to fit the bike).
That said, I’m a bit surprised you’re having significant issues finding a 29x2.35 inch tire. I know for a fact that Schwalbe, Vittoria, and Maxxis all make tires in that size, though they may get a bit pricey.
Once you’ve found a new tire, the tube size just needs to match the size of the tire. Most tubes will safely handle multiple widths of tire, so just look to get a tube that matches the diameter exactly and has the width in it’s range of supported widths.
All that said, the issue at hand may not be the tire, but the rim.
Bicycle tires and rims come either in ‘conventional’ designs, where the tire and rim both have a ‘bead’ on them that locks together when the tube is inflated, or in tubeless designs, which are intended for lower pressures together with a special sealant, that together keep the tire on the rim.
Your current tire is almost certainly a conventional one (it very clearly has a bead), and the fact that it’s slipping out of the rim means that one of four things is the case:
- The rim itself was only designed for tubeless tires. This is potentially expensive to fix, as it means you either need to go tubeless (which requires special preparation of the rim given that you’re using tubes), or get a new wheel or new rim.
- The bead on the rim is damaged in some way, and it’s not locking together with the bead on the tire. This will also require either a new wheel or a new rim.
- The bead on the tire is damaged in some way, and it’s not locking together with the bead on the rim. This can be fixed by replacing the tire, though you should check the rim also, because this type of damage shouldn’t happen outside of a crash (and even in a crash it’s unlikely to happen).
- The tire and/or rim is poorly designed, and you’re running at too low (or too high) of a pressure for them to lock together properly. Replacing the tire may fix this, or it may not.
Regardless of which issue is the case, you should be looking into this in more detail. The tire separating from the rim like that is a potentially dangerous type of failure (at minimum, it greatly increases the chances of getting a flat), and it may indicate other issues with the wheel.