I always get low iron when pg, and my levels aren't so great when not pg. It does sound to me like you're probably iron deficient, but also possibly low on vit b??
I don't think I'd be eating red meat yet because of it, but get a supplement, which would be helpful. I am on ferro grad c(iron with vit c), or fe fol (iron with folate). You could just be exhausted. You do have 3 kids 3 and under. That's tiring. And really even 7hrs sleep most nights, isn't enough rest really (don't worry I know this from experience!). But the weakness and other symptoms aren't normal, get thee to a doctor!!!

Signs of some B vitamin deficiencies:
Vitamin B12 deficiency – this is most common in the elderly, vegans (vitamin B12 is only found in foods of animal origin) and breastfed babies of vegan mothers and is called pernicious anaemia. Symptoms include tiredness and fatigue, lack of appetite and weight loss, apathy and depression, anaemia, smooth tongue and degeneration of peripheral nerves progressing to paralysis.
Folate deficiency – the symptoms include weight loss, tiredness, fatigue and weakness, folate-deficiency anaemia (megaloblastic anaemia) and (during pregnancy) an increased risk of a neural tube defects such as spina bifida for the baby.
Pyridoxine deficiency (B6) – people with excessive alcohol intake, women (especially those on the contraceptive pill), the elderly and people with thyroid disease are at particular risk of deficiency. Symptoms include insomnia, depression, anaemia, smooth tongue and cracked corners of the mouth, irritability, muscle twitching, convulsions, confusion and dermatitis.
Thiamin deficiency – is generally found in countries where the dietary staple is white rice. Deficiencies in the Western world are generally caused by excessive alcohol intake and a very poor diet. Symptoms include confusion and irritability, poor arm or leg (or both) coordination, lethargy, fatigue and muscle weakness.
Nuts, eggs, seeds and legumes (dried beans and lentils) and foods derived from them, such as tofu and hommos, are good sources of protein. You will need one to two serves per day.

Iron deficiency anaemia – Symptoms include looking very pale, breathlessness and fatigue. Reduced immune function and impaired growth and cognition can also be symptoms.
Vegetarians who exclude all animal products from their diet may need almost twice as much dietary iron each day as non-vegetarians. Sources include dark green leafy vegetables – such as spinach or bok choy– and raisins, nuts, seeds, beans, peas, breads and pastas, cereals; especially wholegrain cereals, iron-enriched breakfast cereals and wholemeal breads, Milo and Ovaltine. While eggs, nuts, seeds and legumes also contain iron, it is not as easily absorbed by the body as the iron contained in meat, so vegetarians also need to eat other iron-enriched foods.
Vitamin C increases iron absorption, so eat more brightly coloured fruits and vegetables.
Cut back on the amount of tea and coffee you drink, especially around mealtimes, since the tannins in tea and coffee bind to the iron and interfere with absorption.

Definately go to the doctor though. Even if you get a supplement before you go and can't get to see the doc for a week or two. Could you get a referral from your next antenatal appt?
HTH Melly. Take care of yourself babe. You have 3 1/2 kidlets needing you. If you're not healthy, it makes things twice as hard for you.