Hi and welcome to Pianostreet.
I think one of the problems with 5 finger position playing, not only in sight reading but in general for people new to piano, is that it has a tenancy to constrain focus to a very specific hand position, a very specific fingering, and a very specific set of notes. In reality, the majority of music does not constrain itself in that particular way, to that particular hand position. When someone accustomed to 5 finger position breaks away from it, it can be somewhat of a shock that there are so many other keys and pitches to deal with.
Sight reading fingering is not always the same as performance fingering. When sight reading the focus is to keep pace with the music, using fingerings that allow the pianist to easily maintain their bearing of where they are on the keyboard. Standardized fingerings encountered in scales, chords, arpeggios, and common patterns are often utilized even if they are not the best fit for a passage of music. For example: if you know your fingering for the B major scale, just use it when sight reading scale in that key - there is nothing extra to think about because it is just the same scale you already know, no need to work out a new fingering.
If you run out of fingers, just do what they did in Baroque times when thumb crossing was not in common use, pick up your hand, shift to the new position, continue on. Of course, you should try to not run out of fingers, and this will come with experience.
Read ahead as much as possible. Your eyes should be far ahead in the score from your fingers, that is, you are reading a different part of the score than you are playing. It gives you time to prepare what to do next. Don't stare at a note you are currently playing, it does nothing to aid sight reading, look ahead. Don't stare back at a mistake you made, it doesn't correct the mistake and only distracts you from moving forward, look ahead and fix that spot later. Notes don't change if you look at them longer, read the note and move on, don't dwell on a set of notes for excessively long time. Remember you are sight reading, not doing a deep analysis of the music where you can afford to stop contemplate.
Reading ahead is also what allows you do deal with leaps, because it gives you notice when a leap is coming up, and a chance to prepare your hand.
Practice doing leaps without looking down at the keys. Start with single notes, locating intervals by feel, not by looking, up to an octave. After that you could expand to going beyond an octave to 10ths or 12ths.
Practice jumping around the keyboard, playing a chord in random octaves. Or play a scale, but have each note in a different randomly selected octave.