5.4 is well below normal, and would certainly cause the symptoms you describe, and be justification for hormone replacement therapy.
The questions going through your doctor's mind at the moment is whether there is any direct cause, such as malfunction or tumours in the endocrine system, or whether its just a natural decline with age. Unfortunately the latter is difficult to establish without records of what your 'normal' testosterone levels were say, in your twenties versus thirties versus now, but if everything else is ruled out then this becomes the 'default' diagnosis.
There's also growing opinion that declining testosterone with age is not something we should naturally accept, and that even those within the 'normal' range but with declining levels can benefit from supplements. Testosterone affects so many things we associate with a healthy active lifestyle (energy levels, concentration, muscle strength, drive and ambition, sex!), that you've got to question why anyone would accept losing this as they get older if the treatment is readily available and has very few downsides - both of which seem to be true!
Of course we're not talking about taking levels which count as abuse which some athletes and body builders go to, just returning levels to 'normal', and it is after all a natural substance already in the body. So, I wouldn't worry about the prospect of supplementing it - its simple, painless, and effective! The only real downside is increased risk of prostate cancer, so you'll need to be regularly tested for that.
Once the cause of your low levels has been established, and if supplements are deemed to be the solution, then you'll likely be faced with a number of choices for actually administering the stuff. There's tablets, but ingesting testosterone isn't the favoured solution these days. Adsorption through the skin thru gels or patches, or injection of a slowly dissipating gel or pellet into a muscle are the more usual routes. The gels and patches are daily and the dose can be modified quite easily, whereas injections are monthly or even less frequently so are less hassle, but you have to go to the doctor each time.
I was diagnosed with low testosterone a couple of months back, and have been on the patches since then. Unfortunately my skin is starting to react to them and I may have to find another method soon. In the meantime the effects have been quite marked and immediate - I've regained my mojo!
But, as always, my advice is just from someone you don't know on a forum, everyone's situation is different, and follow your doctor's advice rather than mine.